Contents

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The N-fold Darboux transformations andexact solutions of the Boussinesq-Burgersequation

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10:50-11:10 A Note on Two Classes of Boolean Func-tions With Optimal Algebraic Immunity

Analytical nonautonomous matter-wavesolutions in Bose-Einstein condensate

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11:10-11:30 The Optimal Linear Secret SharingScheme for Any Given Access Structure

An algorithm for decision of the existenceof nonclassical symmetry

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10:00-10:20 An Efficient Method of Computing Com-prehensive Grobner Bases and MinimalComprehensive Grobner Bases

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Detecting Strong Nontermination of Multi-Path Polynomial Programs

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10:50-11:10 Termination of Zeilberger-style algorithms:the mixed differential-q-shift and shift-q-shift cases

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Primary Decomposition of Polynomial Ideals

pXöClemson University, {I

The primary decomposition theorem, established by Lasker (1905) andNoether (1921), is a milestone in commutative algebra. It traces back toantiques in number theory in terms of integer and polynomial factorizationand is a key witness to the development of modern algebra. Algorithmsfor computing primary decomposition have been studied extensively since1920s and some of them are implemented in major computer algebra systems(e.g. Maple, Magma and Mathematica). However, efficient computation ofprimary decomposition is still a major challenge today even for intermediatesize of polynomial systems (note that it is NP hard in general). In thistalk, I shall give a brief survey of the basic ideas for computing primarydecomposition, including a recent algorithm of the speaker with DaqingWan and Mingsheng Wang on 0-dimensional ideals over finite fields.

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An algorithm for decision ofthe existence of nonclassical symmetry

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An algorithm which uses differential characteristic set algorithm for thedecision of the existence of nonclassical symmetry of partial differential equa-tions is proposed. The algorithm partially gives answer for an open problemproposed by P.A. Clarkson on nonclassical symmetry of partial differentialequations. As applications of our algorithm, several examples of determiningthe nonclassical symmetries of some evolution equations are presented.

On the Structure of Multivariate Mixed q-Hypergeometric Terms

��«1, GI¹2, ¾X]2, x§2

1Department of Mathematics, North Carolina State University2Key Lab of Math.-Mech., Chinese Academy of Sciences

schen21@ncsu.edu(Shaoshi Chen), ryfeng@amss.ac.cn(Ruyong Feng),fuguofeng@mmrc.iss.ac.cn(Guofeng Fu), jinkang@mmrc.iss.ac.cn(Jin Kang)

Hypergeometric terms and their q-analogue have played an importantrole in the study of special functions. In combinatorics, a large class of enu-merative problems are encoded by sums or identities of those terms. In theearly 1990’s, Zeilberger formulated a method, namely creative telescoping,which has become a powerful tool for automatically proving many combina-torial identities involving integrals or sums of hyperexponential functions,hypergeometric terms and their q-analogue [2, 6, 8].

One of the challenging problems related to Zeilberger’s method is thetermination problem. For bivariate hypergeometric terms, Abramov [1] ob-tained a termination criterion for Zeilberger’s algorithm. Later, Abramov’scriterion was extended to the cases of bivariate q-hypergeometric terms [5]and mixed hyperexponential-hypergeometric functions [3]. All of those re-sults were based on structure theorems of the bivariate inputs. In orderto prove a conjecture by Wilf and Zeilberger [6], Chen and Li presented astructure theorem for multivariate hyperexponential-hypergeometric func-tions in [5]. Recently, a more general structure theorem has been derived

7

in [4], where the functions may involve several q-variables. In this pa-per, we will present a refined version of the structure theorem for mixedq-hypergeometric terms.

References

[1] S. A. Abramov. When does Zeilberger’s algorithm succeed? Adv. inAppl. Math., 30(3):424–441, 2003.

[2] G. Almkvist and D. Zeilberger. The method of differentiating underthe integral sign. J. Symbolic Comput., 10:571–591, 1990.

[3] S. Chen, F. Chyzak, R. Feng, and Z. Li. The existence of telescopers forhyperexponential-hypergeometric functions, 2010. MM-Res. Preprints(2010) No. 29, 239-267.

[4] S. Chen, R. Feng, G. Fu, and Z. Li. On the Structure of Compatible Ra-tional Functions. In ISSAC ’11: Proceedings of the 2011 InternationalSymposium on Symbolic and Algebraic Computation.

[5] S. Chen and Z. Li. A multiplicative form of multivariatehyperexponential-hypergeometric functions, 2010. MM-Res. Preprints(2010) No. 29, 25-35.

[6] W. Y. C. Chen, Q.-H. Hou, and Y.-P. Mu. Applicability of the q-analogue of Zeilberger’s algorithm. J. Symbolic Comput., 39(2):155–170, 2005.

[7] H. S. Wilf and D. Zeilberger. An algorithmic proof theory for hypergeo-metric (ordinary and “q”) multisum/integral identities. Invent. Math.,108(3):575–633, 1992.

[8] D. Zeilberger. The method of creative telescoping. J. Symbolic Comput.,11(3):195–204, 1991.

Termination of Zeilberger-style algorithms:the mixed differential-q-shift and shift-q-shift cases

��«1, GI¹2

1Department of Mathematics, North Carolina State University2Key Lab of Math.-Mech., Chinese Academy of Sciences

schen21@ncsu.edu fuguofeng@mmrc.iss.ac.cn

8

Creative telescoping introduced by Zeilberger [7, 8] is an important toolfor algorithmic integration and summation of special functions. Severalresults of the last two decades are fundamental for deciding the terminationof algorithms based on it. Wilf and Zeilberger have shown that telescopersalways exist for proper hypergeometric functions [6]. In the bivariate discretecase, Abramov has proved that a hypergeometric function can be written asa sum of a hypergeometric-summable function and a proper one if it has atelescoper [1]. Similar results have (more recently) been obtained in the q-shift case by B. Chen et al [5]. In the bivariate continuous-discrete case, thesimilar result has been given by S. Chen et al [3]. We consider the bivariatedifferential-q-shift case and shift-q-shift case, where q is not a root of unity.

We present two criteria on the termination of Zeilberger’s algorithmin the mixed cases. The first is for the differential and q-shift case; thesecond is for the shift and q-shift case. The criteria describe necessary andsufficient conditions on the existence of telescopers for the hyperexponetial-hypergeometric solutions in the above mixed cases.

Let k be a field of characteristic zero and k(t, y) be the field of rationalfunctions in t and y over k. For every f ∈ k(t, y), we define

Dt(f) =∂f

∂tand Sy(f(t, y)) = f(t, qy),

where q is not a root of unity in k. We denote by k(t, y)〈Dt, Sy〉 the ring ofdifferential-q-shift operators with rational function coefficients in k(t, y). Afirst order differential-q-shift system

Dt(z) = a z, Sy(z) = b z, with a, b ∈ k(t, y) (1)

is said to be compatible if b 6= 0 and Dt(b)/b = Sy(a) − a. Accordingto Theorem 2 in [2], there always exists a simple differential-q-differenceextension A such that the compatible system (1) has a nonzero solutionin A. The solution space of system (1) in A is denoted byH(a, b). We call anelement in H(a, b) a hyperexponential function over k(t, y) with certificates aand b. By [4, Lemma 3.2], a hyperexponential function h(t, y) can be writtenin the form:

h(t, y) = f(t, y) · E(t) · Q(y),

where f ∈ k(t, y), E(t) is a hyperexponential function with respect t andQ(y) is a hyperexponential function with respect y. A hyperexponentialfunction is said to be proper if f is a polynomial in k[t, y].

Problem 1 For a nonzero hyperexponential function h(t, y) over k(t, y),

9

(i) decide whether there exists a nonzero operator L(y, Sy) in k(y)〈Sy〉 s.t.

L(y, Sy)(h) = Dt(g), (2)

for some hyperexponential function g. Such L is called a telescoperw.r.t. t.

(ii) decide whether there exists a nonzero operator L(t,Dt) in k(t)〈Dt〉 s.t.

L(t,Dt)(h) = (Sy − 1)(g), (3)

for some hyperexponential function g.Such L is called a telescoperw.r.t. y.

First, we recall the definition of additive decomposition.

Definition 2 (Additive decomposition with respect to t) Let h be in H(a, b)with a, b ∈ k(t, y). An additive decomposition of h with respect to t is of theform

h = Dt(h1) + h2,

where h1 is hyperexponential over F(t), and either h2 is zero or

h2 ∈ u(t, y) · H(a2, b2), where b2 ∈ k(t, y) and (u, a2) satisfies

(i) den(u) is square-free with respect to t;

(ii) a2 is differential reduced with respect to t;

Definition 3 (Additive decomposition with respect to y) Let h be in H(a, b)with a, b ∈ k(t, y). An additive decomposition of h with respect to y is of theform

h = (Sy − 1)(h1) + h2,

where h1 is hyperexponetial over k(t, y), and either h2 is zero or

h2 ∈ u(t, y) · H(a2, b2), where a2 ∈ k(t, y) and (u, b2) satisfies

(i) den(u) is q-shift free with respect to y;

(ii) b2 is q-shift reduced with respect to y;

Note that there are additive decompositions of various kinds. The aboveones appear to be the most coarse one.

We are going to present our ideas to prove the following conclusion.

10

Conclusion 1 Let h(t, y) be a hyperexponential function over k(t, y).

(i) If h = Dt(h1) + h2 is an additive decomposition of h with respect to t.Then h has a telescoper with respect to t if and only if h2 is proper.

(ii) If h = (Sy − 1)(h1) + h2 is an additive decomposition of h with respectto y. Then h has a telescoper with respect to y if and only if h2 isproper.

In the bivariate shift-q-shift case, in order to describe the problem in aunified form, let z1, z2 be in {x, y} such that z1 6= z2 and θ1, θ2 be in {Sx, Sy}such that θi(zi) = zi for i = 1, 2.

Problem 4 For a nonzero hypergeometric term h(z1, z2) over k(z1, z2), de-cide whether there exists a nonzero operator L(z1, θ1) in k(z1)〈θ1〉 such that

L(z1, θ1)(h) = (θ2 − 1)(g), (4)

for some hypergeometric term g. Such an operator L(z1, θ1) is called a tele-scoper for h with respect to z2.

Similar to the differential-q-shift case, we present our ideas to prove thefollowing conclusion.

Conclusion 2 Let h(z1, z2) be a nonzero hypergeometric term over k(z1, z2).If h = (θ1− 1)(h1) +h2 is an additive decomposition of h with respect to z1.Then h has a telescoper with respect to z1 if and only if h2 is proper.

References

[1] S.A. Abramov. When does Zeilberger’s algorithm succeed? Adv. inAppl. Math. , 30(3):424–441, 2003.

[2] M. Bronstein, Z Li and M. Wu. Picard–Vessiot extensions for linearfunctional systems, Proceedings of ISSAC’05 68–75, 2005.

[3] S.Chen, F.Chyzak, R.Feng and Z.Li. The existence of telescopers forhyperexponential-hypergeomeytric functions, MM-Res.Preprints, No.29,239-267, 2010.

[4] S.Chen, R.Feng, G.Fu and Z.Li On the structure of compatible rationalfunctions, Proceeding of ISSAC’11, 91–98, 2011.

11

[5] Y.C. Chen, Q. Hou and Y. Mu. Applicability of the q-analogue of Zeil-berger’s algorithm, J. Symbolic Comput., 39(2):155–170, 2005.

[6] H. Wilf and D. Zeilberger An algorithmic proof theory for hypergeo-metric (ordinary and “q”) multisum/integral identities . Invent. Math.,108(3):575–633, 1992.

[7] D. Zeilberger. A holonomic systems approach to special functions iden-tities. J. Comput. Appl. Math., 32:321–368, 1990.

[8] D. Zeilberger. The method of creative telescoping. J. Symbolic Comput.,11(3):195–204, 1991.

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References

[1] P.Paule,C.Schneider Truncating binomial series with symbolic sum-mation Integers:Electronic Journal of Combinatorial Number Theory7:#A22, 2007

An Algorithmic Approach to the Center Conditions forPolynomial Lienard Systems

Yizheng Hua, Û]b

aInstitute of Computer Applications, Academia Sinica, Chengdu 610041,China

bCollege of Mathematics and Information Science, Wenzhou University,Wenzhou 325035, China

Motivated by the development of computer and computer algebraic al-gorithm, there are a lot of literatures considering the qualitative property ofdifferential polynomial systems during the last decades, especially the cen-ter problem of differential polynomial systems (see [10] and the referencetherein). As one of the most well known differential systems, Lienard sys-tems have been studied in many aspects. Connected to the Hilbert’s 16thproblem, the Hopf cyclicity problem of Lienard system has been consideredin a series of papers of N.G. Lloyd[4] and M. Han[12]. The center problemis another aspect attracted the interest of many researchers [3, 6, 11].

The so-called Lienard system is in the form,

x = y,y = −g(x)− yf(x).

(1)

When f(x) and g(x) are polynomials, it is called the polynomial Lienardsystem.

Let F (x) =∫ x0 f( ∂

∂xi)d ∂

∂xi, G(x) =

∫ x0 g( ∂

∂xi)d ∂

∂xi. The following theo-

rem is by Cherkas’s method from [3], see also[5, 8].

Theorem 0.1 System (1) has a center at the origin if and only if thereexists a real analytic function z(x), defined in a neighborhood of the origin,with z(0) = 0 and z′(0) = −1, satisfying

F (x) = F (z(x)), G(x) = G(z(x)).

13

With the Luroth.s theorem, the center problem of polynomial Lienardsystems is converted to the generator construction problem in some subfieldof the univariate rational function field. In [5], C. Christopher had con-sidered the algebraic center conditions for polynomial Lienard systems andthey used these conditions to give a simple classification of such centers.From a theorem of Fried and MeRac[7], the problem can also be changed tothe problem of polynomial decomposition[9]. In a different way, we derivethe same result.

Theorem 0.2 The system (1) with g(0) = 0 and g′(0) > 0 has a nondegen-erate center at the origin if and only if F (x) and G(x) are both polynomialsof a polynomial h(x) with h′(0) = 0 and h′′(0) 6= 0.

Different from Christopher, we emphasize the algorithmic approach tothe discrimination of centers. Based on an algorithm of polynomial decomposition[1,2], we give a simple algorithm to check the center problem for polynomialLienard systems and some concrete examples are illustrated.

References

[1] F. Binder, Polynomial decomposition. Master’s thesis, University ofLinz, June 1995.

[2] F. Binder. Fast computations in the lattice of polynomial rational func-tion fields. In Proc. ISSAC-96. ACM Press (1996), 43-48.

[3] L. A. Cherkas, Conditions for a Lienard equation to have a center,Differential Equations 12(1977), 201õ206.

[4] C.J. Christopher, N.G. Lloyd, Small-amplitude limit cycles in polyno-mial Lienard systems, Nonlinear Differential Equations Appl. 3 (1996),183-190.

[5] C. Christopher, An algebraic approach to the classification of centersin polynomial Licenard systems, J. Math. Anal. Appl. 229 (1999),319õ329.

[6] C. Christopher, D.Schlomiuk, Center conditions for a class of polyno-mial differential system, CRM-3236,(2006).

[7] M. Fried, R. MacRae, On curves with separated variables. Math. Ann.10 (1969), 220-226.

14

[8] A. Gasull, J. Torregrosa, Center problem for several differential equa-tions via Cherkas’ method, J. Math. Anal. Appl., 228 (1998), 322 - 343.

[9] J. von zur Gathen, Functional decomposition of polynomials: the tamecase. Journal of Symbolic Computation 9 (1990), 281-299.

[10] V.G. Romanovski and D.S. Shafer, The Center and Cyclicity Problems:A Computational Algebra Approach, Birkhauser, Boston, 2009.

[11] Q. Xu, W. Huang, The center conditions and local bifurcation of criticalperiods for a Lienard system, Applied Mathematics and Computation217 (2011) 6637-6643.

[12] J. Yang, M. Han, V. G. Romanovski, Limit cycle bifurcations of someLiWnard systems , J. Math. Anal. Appl. 366 (2010) 242-255.

(Supported by the National Natural Science Foundation of China (GrantNo. 11001204))

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ëëë���©©©zzz:[1] Dunbar.S§Travelling wave in diffusive Lotka-Volterra equations[J]§Journal of Biol-

ogy§1983.17:11-32"[2] Dunbar.S§Travelling wave solutions of diffusive predatorprey equations:periodic or-

bits and point to periodic hertocline orbits[J]§SIAM J Appl Math§1986.46:1057-1078"

[3] J.Huang G.Lu S.Ruan§Existence of travelling wave solutions in a diffusive preda-torprey model[J].J Math Biol§2003.46:132-153.

[4] W.Li S.Wu§Travelling waves in a diffusive predatorprey model with holling type-IIIfunctional response[J]§Chaos.Solitons and Fractals§2008.37:476-487"

[5] �Ú§AaHolling.Ó - �XÚ�1Å)�ïÄ[D]§§²�Æa¬Ø©§2010.5"

[6] J.T.Tanner§The stability and the intrinsic growth rates of prey and predator popu-lations[J]§Ecology§1975.56:855-867"

[7] Sze-Bi Hsu Tzy-Wei Huang§Global stability for a class of predatorprey sys-tems[J]§SIAM J Appl Math§1995.55:763-783"

[8] Wonlyul Ko Kimun Ryu§Non-constant positive steady-states of a diffusive predator-prey system in homogeneous environment[J]§Math Anal Appl§2006.327:539-549"

A general conjecture similar toT-D conjecture and its applications in

constructing Boolean functions with optimal algebraic immunity

@��!4R�!Ç�¸!Ü¡²Key Laboratory of Mathematics Mechanization

Institute of Systems Science, AMSSBeijing 100190, China

Boolean functions, which are used in the combiner and filter models of streamciphers and for S-box designing in block ciphers, play an critical role in symmetriccryptographic systems. Boolean functions are generally required to be balancedand have high algebraic degree, high nonlinearity, high correlation immunity andhigh algebraic immunity. In this paper, we propose two constructions of 2k-variableBoolean functions.

Construction 5 Let n = 2k ≥ 4, (u, 2k − 1) = 1. Let α be a primitive element of

the finite field F2k . Set ∆s = {αs, αs+1, · · · , α2k−1+s−1} where 0 ≤ s < 2k− 1 is aninteger. Then we define a function f ∈ Bn as follows

f(x, y) = g(xy2k−1−u),

where g is a Boolean function defined over F2k with Supp(g) = ∆s.

18

Construction 6 Let n = 2k be an even integer, k ≥ 2. Let α be a primitive

element of the finite field F2k . Set ∆s = {αs, · · · , α2k−1+s−1} where 0 ≤ s < 2k − 1is an integer. We define the Boolean F ∈ Bn as follows

F (x, y) =

{g(xy2k−1−u), x 6= 0;g(y), x = 0.

where g is a Boolean function defined on F2k with supp(g) = ∆s.

These classes of Boolean functions have optimal algebraic immunity under theassumption that a general combinatorial conjecture is correct. These functions alsohave high algebraic degree and high nonlinearity. Boolean functions defined inConstruction 1 contain more bent functions. Boolean function in Construction 2are balanced.

A Note on Two Classes ofBoolean Functions With Optimal Algebraic Immunity

@��!4R�!Ç�¸!Ü¡²KLMM, Academy of Mathematics and Systems Science

Chinese Academy of Sciences

Tu and Deng proposed a class of bent functions which are of optimal algebraicimmunity under the assumption of a combinatorial conjecture. They are the firstto find the connection between bentness and algebraic immunity of Boolean func-tions. In this note, we compute the dual of Tu-Deng functions and then show thatthey are still of optimal algebraic immunity under the assumption of the same con-jecture. For another class of Boolean functions constructed by Tang et al whichare algebraic immunity optimal based on a conjecture similar to Tu and Deng’s,we show that they are not bent functions by using some basic properties of binarycomplete Kloosterman sums.

Keywords Boolean function, Algebraic immunity, Bent function, Walsh trans-form, Kloosterman sums.

Minimum time trajectory planning forfive-axis machining with general kinematic constraints

oäJ1 Ür1 p�ì2

1College of Information and Control Engineering, China University of Petroleum(East China), Qingdao, China

2KLMM, Institute of Systems Science, Chinese Academy of Sciences, Beijing,China

19

keyword: Time optimal, Trajectory planning, Five-axis machining, CNCFive-axis machines are wildly used in manufacturing industry at present. In

comparison with three-axis machining, the five-axis machining has the advantagesof reducing machining time, eliminating multiple setups, and machining complexsurfaces. The problem of planning time optimal feedrate trajectory for five-axismachining has received wide attention in recent years.

In this paper, an optimization approach is proposed for generating smooth andtime-optimal trajectory for five-axis computer numerical control (CNC) machining.The desired feedrate trajectory planning (FTP) problem is formulated as an equiv-alent optimal control problem. The velocity, acceleration and jerk limits of the fivedrives are considered in the problem. The desired smoothness of the trajectory canbe obtained by adjusting the values of jerk limits. Since of the complex nonlinearmapping between the tool pose (tool tip position and tool axis orientation) and theactual drive axis positions, it is hard to control the cutting feedrate in the workpiececoordinate system (WCS) under the condition of only limiting the drives kinematicsperformance. And it does not guarantee the machining accuracy. Thus the chorderror limit of the tool tip motion is also considered in order to limit the cuttingfeedrate. The time optimal trajectory is obtained by solving the equivalent optimalcontrol problem using a control vector parameterization (CVP) method. A detailedproof is provided to shown that the optimal feed has “bang-bang” structure, i.e. atleast one of the five axes reaches its limit throughout the motion. The effectivenessof the approach is demonstrated in machining simulation of a jet engine impeller.

Estimating defeaturing induced engineering analysis error

o², pܲState Key Laboratory of CAD&CG, Zhejiang University, China P.R.

The reports summarizes our recent results (published or unpublished) on es-timating defeaturing induced engineering analysis error. This is a critical issueinvolved in seamless CAD/CAE integration for converting an original design solidmodel to an analysis-suitable analysis solid model for downstream tasks of meshingand running simulation.

Computer simulations, or engineering analysis, of physical events is generallyperformed using finite element (FE) analysis on a volumetric mesh derived by dis-cretizing a continuous geometric model. Typically, a process of model simplificationis required prior to mesh generation to remove irrelevant geometric details that havelittle impact on the results of analysis, allowing the analysis to be performed morequickly and with lower memory requirements on a simpler model. Precisely, thepresence of these irrelevant geometric details can significantly increase the time andcomputational complexity both for the meshing process and the FE analysis per-formed on it. In the worse case they may even lead to mesh generation failure orill-conditioned computations that may produce inaccurate analysis results. Simply

20

using greater computing power may not be adequate to resolve these issues, andmodel simplification is essential in such cases.

Two major sources of errors in engineering analysis of physical events weregenerally studied in previous research studies: approximation error, due to theinherent inaccuracies incurred in the discretization of mathematical models of theevents, and modeling error, due to the natural imperfections in abstract models ofactual physical phenomena. In these studies, the underlying geometry over whichanalysis is performed, and the associated boundary conditions, remains unchangedbefore and after approximation. The problem of estimating analysis error inducedby model simplification, where the underlying geometry is changed, is howeverrarely studied.

In order to handle this issue, we develop a general framework of estimating mod-eling error caused by geometric model simplification for general physical events. Thesimplification is performed by removing from a complex geometric model explicitlygiven negative features that may be loaded with prescribed Neumann conditions.The error is measured in terms of changes of specific quantities of engineering inter-est, and so is a goal-oriented error. It is evaluated in an a posteriori sense by usingsolutions of the simplified model but not solutions of the original complex model.

The proposed approach is based on the idea that the simplification error causedby geometric differences can be taken as a modeling error caused by different mathe-matical modeling defined over the same geometric model. Estimating this modelingerror is achieved using the dual weighted residual (DWR) method, originally de-veloped in by Becker et al and extended by Oden, in combination with an exteriorapproximation for solution differences in linear cases. By further simplificationand estimation, ultimately, the error estimator is expressed in the form of a localintegration over the feature’s boundary, which can be explicitly evaluated usingsolutions of the defeatured model.

Performance of the estimator is also tested for various engineering analysis prob-lems, including Poisson equation, linear elasticity problem and nonlinear diffusion-reaction equation. Results were also compared with results obtained by classicaltopological sensitivity analysis.

Sparse Differential Resultant forLaurent Differential Polynomials

o�KLMM, Academy of Mathematics and Systems Science

Chinese Academy of Sciences

In this talk, we introduce the concepts of Laurent differential polynomials andLaurent differentially essential systems and give a criterion for a system to be aLaurent differentially essential one in terms of its support. We then define thesparse differential resultant for a Laurent differentially essential system and discussits basic properties, which are similar to those of the algebraic sparse resultant.

21

In particular, we obtain the order and degree bounds for the sparse differentialresultant. Based on these bounds, we propose an algorithm to compute the sparsedifferential resultant. The algorithm is single exponential in terms of the order, thenumber of variables, and the size of the Laurent differential system.

This is a recent joint work with my advisor Xiao-Shan Gao, and Dr. Chun-MingYuan.

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Detecting Strong Nontermination ofMulti-Path Polynomial Programs

44ô!M´!ÉD�!ëð�State Key Laboratory of Computer Science, Institute of Software,

Chinese Academy of Sciences, Beijing 100190, P.R. China{liuj,xum,znj,zhaohj}@ios.ac.cn

22

Automatic discovery of termination proofs for programs is significant in pro-gram verification. However, this problem generally is equivalent to the famoushalting problem, and is therefore undecidable. Hence detecting nontermination asa complementary of proving termination is necessary in practice. In this paper, weconsider the model of multi-path polynomial programs, and propose the notion ofstrong nontermination to under-approximate nontermination. Based on polynomialideal theory, a sufficient and necessary criterion for strong nontermination is for-mulated. As a result, the set of all strongly nonterminating inputs are computable.

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Grobner bases with respect toseveral orderings on difference-differential modules and

multivariate dimension polynomials

4==1,2 ±�2

1Guizhou university for nationalites, Guiyang(550025), Guizhou, China2School of Mathematics and System Science and LMIB, Beihang University,

Beijing(100191), Chinazhoumeng1613@hotmail.com

In [Lev07] the Grobner basis method was generalized to the case of free modulesover Ore polynomial rings with respect to several term orderings associated witha partition of the set of variables. Levin considered an Ore polynomial ring D asa filtered ring with the natural p-dimensional filtration and introduced a specialtype of reduction in a free D-module and develop the corresponding Grobner basistechnique. With the technique the difference-differential dimension polynomialsin several variables can be computed. But in [Lev07] the difference operators aresupposed non-inversive. In [ZhW08] the authors introduced a concept of relativedifference-differential Grobner bases, for algorithmically computing the difference-differential dimension polynomials in two variables and the difference operators areinversive. The notion of relative Grobner basis is based on two generalized termorders on Nm×Zn. They define a special type of reduction for two generalized termorders in a free left module over a ring of difference-differential operators. Thenthe concept of relative Grobner bases was introduced.

In this paper we present a new algorithmic approach for computing the Grobnerbases with respect to several generalized term orders on Nm×Zn and on difference-differential modules. We define a special type of reduction for several generalizedterm orders in a free left module over a ring of difference-differential operators. Thisreduction is different from the reduction in [Lev07]. Then the concept of Grobnerbases with respect to several generalized term orders is defined. An algorithm forconstructing these Grobner bases is presented and verified. Using the Grobnerbases, we are able to compute difference-differential dimension polynomials in sev-eral variables in the case of that the difference operators are inversive. So our resultshave improved the theorem of [Lev07] where Levin considered difference-differentialdimension polynomials in several variables for modules over Ore polynomial ringswith non-inversive difference operators. Also, our result is a generalization of the-ories of [ZhW08] and include them as special cases.

Let R be a commutative noetherian ring, ∆ = {δ1, · · · , δm} a set of deriva-tions and Σ = {σ1, · · · , σn} a set of automorphisms of the ring R, which commutewith each other; i.e. α ◦ β = β ◦ α for all α, β ∈ ∆ ∪ Σ. Then R is called adifference − differential ring with the basic set of derivations ∆ and the basicset of automorphisms Σ, or shortly a ∆− Σ-ring. If R is a field, then it is called a∆− Σ-field.

24

In this paper we suppose that R is a ∆− Σ-field and

D = R[δ1, · · · , δm, σ1, · · · , σn].

Elements of D are of the form∑λ∈Λ aλλ, λ = δk11 · · · δkmm σl11 · · ·σlnn , where (k1, · · · , km) ∈

Nm and (l1, · · · , ln) ∈ Zn.Note that δa = aδ + δ(a), σa = σ(a)σ for all a∈ R, δ ∈ ∆, σ ∈ Σ. The terms

λ ∈ Λ do not commute with the coefficients aλ ∈ R. A left D-module M is calleda difference-differential module. If M is finitely generated as a left D-module, thenM is called a finitely generated ∆− Σ-module.

Let F be a finitely generated free D-module with a set of free generators E ={e1, · · · , eq}. Then F can be considered as an R-module generated by the set ofall elements of the form λei(i = 1, · · · , q, ) . If ”≺” is a generalized term order (cf[ZhW06]) on Nm × Zn × E then ” ≺ ” induces a generalized term order on ΛE.

Every element f ∈ F has a unique representation as a linear combination ofterms:

f = a1λ1ej1 + · · ·+ adλdejd

The main idea and results are following.Definition 3.2. Given f, g, h ∈ F , with g 6= 0, we say that the element f

modulo g reduces to h with respect to ≺1, · · · ,≺k in one step and write fg−−−−−−→

≺1,··· ,≺k

h, if f contains some term w with a coefficient a and there exist a term λ ∈ Λ suchthat w = lt≺1

(λg),

h = f − a

lc≺1(λg)

(λg)

and lt≺j (λg) �j lt≺j (f), j = (2, · · · , k)Definition 3.3. Let f, h ∈ F and let G = {g1, · · · , gp} be a finite set of non-zero

elements of F. We say that f reduces to h modulo G with respect to ≺1, · · · ,≺k andif and only if there exists a sequence of elements gi1 , · · · , giq ∈ G and a sequence ofelements h1, · · · , hq−1 ∈ E such that

fgi1−−−−−−→

≺1,··· ,≺k

h1

gi2−−−−−−→≺1,··· ,≺k

· · ·giq−1−−−−−−→≺1,··· ,≺k

hq−1gq−−−−−−→

≺1,··· ,≺k

h

If h cannot reduce modulo G with respect to ≺1, · · · ,≺k we say h is reduced.Theorem 3.1. Let ≺1, · · · ,≺k are term orders on ΛE. Let g1, · · · , gp ∈ F \{0}

and f ∈ F . Then exist r ∈ F, h1, · · · , hp ∈ D, such that

f = h1g1 + · · ·+ hpgp + r

and r is reduced with respect to ≺1, · · · ,≺kDefinition 3.4. Let ≺1, · · · ,≺k are generalized term orders on ΛE. Let G =

{g1, · · · , gp} ∈W \ {0}. If for every f ∈W , f can be reduced to 0 modulo G withrespect to ≺1, · · · ,≺k. Then G is a Grobner basis of W with respect to ≺1, · · · ,≺k.

25

Definition 3.5. Let F be a finitely generated free D-module and f, g ∈ F{0}.Let ≺r be a generalized term order on ΛE. For every Λj let V (j, f, g) be a finitesystem of generators of the R[Λj ]-module

R[Λj ]〈lt≺r(λf) ∈ ΛjE|λ ∈ Λ〉 ∩R[Λj ] 〈lt≺r

(ηg) ∈ ΛjE|η ∈ Λ〉

Then for every generator v ∈ V (j, f, g)

S≺r (j, f, g, v) =v

(lt≺r)j(f)

f

lc(≺r)j(f)

− v

lt(≺r)j(g)

g

lc(≺r)j(g)

is called an S≺r-polynomial of f and g with respect to j, r and v.

Theorem 3.3. Let F be a free D-module, ≺1, · · · ,≺k be generalized term orderson ΛE, G be a finite subset of F \ {0} and W be the submodule in F generated byG. Then for ever r < k, (r = 1, · · · , k − 1), G is a Grobner basis of W with respectto ≺r, · · · ,≺k if and only if G is a Grobner basis of W with respect to ≺r+1, · · · ,≺kand for all Λj , for all gi, gk ∈ G and for all v ∈ V (j, gi, gk), the S≺r

-polynomialsS≺r

(j, gi, gk, v) can be reduced to 0 modulo G with respect to ≺r, · · · ,≺k.Let R be a ∆ − Σ-field, D the ring of ∆ − Σ-operators over R, M a finitely

generated ∆ − Σ- module (i.e. a finitely generated difference-differential module),F a finitely generated free ∆ − Σ module. We will continue to use the notationsand conventions of the preceding sections.

As an application, we have the following theorem.Theorem 4.1. Let R be a ∆ − Σ-field, M be a finitely generated D-module

with generators h1, · · · , hq . Let F be a free ∆− Σ module with a basis e1, · · · , eqand π : F → M the natural ∆ − Σ-epimorphism of F onto M (π(ei) = hi for1 = 1, · · · , q). Let ≺,≺1, · · · ,≺k be the generalized term orders on ΛE of theterms of F. Consider the submodule N = kerπ of F and let G = {g1, · · · , gp}be a Grobner basis of N with respect to ≺,≺1, · · · ,≺k. Furthermore, for any(r, r1, · · · , rk) ∈ Nk+1. Let

Urr1···rk = {w ∈ ΛE| |w| ≤ r, |w|j ≤ rj , w 6= lt≺(λgi) for all λ ∈ Λj , g ∈ G}⋃{w ∈ ΛE| |w| ≤ r, |w|j ≤ rj , and there exits t ∈ {1, · · · , k}, |lt≺t

(λgi)|t > rt

for all λ ∈ Λj , g ∈ G s.t. w = lt≺(λgi)}

Then for any r, r1, · · · , rk ∈ Nk+1 the set Urr1···rk is a R-basis of Mrr1···rk , and themultivariate difference-differential dimension polynomial ψ(t, t1, · · · , tk) associatedwith M with respect to the p-dimension filtration of M is the cardinality of Urr1···rk ,i.e.

ψ(t, t1, · · · , tk) =∣∣Urr1···rk ∣∣.

References

[Lev07] Levin, A.B. (2007). Grobner bases with respect to several orderings andmultivariable dimension polynomials. J.Symb. Comput., 42/5, 561–578.

26

[ZhW06] Zhou, M., Winkler, F. (2006). Grobner bases in difference-differentialmodules. In Proceedings ISSAC 2006, 353-360. ACM Press.

[ZhW08] Zhou, M., Winkler, F. (2008). Computing difference-differential dimen-sion polynomials by relative Grobner bases in difference-differential modules.J. Symb. Comput., 43, 726-745.

A Signature-Based Algorithm for Computing Grobner Basis inQuasi-Commutative Rings

ê¡Å!��!�½xKLMM, Academy of Mathematics and Systems Science, CAS, Beijing 100190,

China(maxiaodong, sunyao)@amss.ac.cn, dwang@mmrc.iss.ac.cn

Let k be a field, N be the set of non-negative integers, andR := k < x1, · · · , xn >be a finitely generated algebra over k, i.e. any f ∈ R has a form of f = Σα∈Nncαx

α

where cα ∈ k, xα = xa11 · · ·xann and α = (a1, · · · , an). Let ≺ be an admissible orderon Nn, i.e. a total order on Nn such that 0 ∈ Nn is the smallest element and α ≺ βimplies α+ γ ≺ β + γ for all α, β, γ ∈ Nn. Then for 0 6= f =

∑α∈Nn cαx

α ∈ R, thedegree of f is defined as deg(f) := max≺{α | cα 6= 0} ∈ Nn.

The algebra R is called a quasi-commutative ring, if R is a ring and for anyf, g ∈ R, the relations

fg − gf = 0 or deg(fg − gf) ≺ deg(fg) = deg(gf)

always hold. There are two important kinds of quasi-commutative rings: polyno-mial rings and Weyl algebras. In a polynomial ring Rpoly = k[x1, · · · , xn], any twopolynomials f, g ∈ Rpoly have the relation fg−gf = 0, since Rpoly is commutative.In a Weyl algebra RWeyl = k[x1, · · · , xn, D1, · · · , Dn] where Di = ∂

∂xiis the partial

derivative by xi for i = 1, · · · , n, the following relations hold: xixj − xjxi = 0,DiDj −DjDi = 0, Dixj − xjDi = 0 and Dixi − xiDi = 1 for any i 6= j.

The Grobner basis for the quasi-commutative ring R is defined as follows:

Definition 7 Let J be an ideal in R and G a finite subset of J \ {0}. Then G isa Grobner basis of J w.r.t. ≺, iff for all f ∈ J , there exists g ∈ G such that

deg(f)− deg(g) ∈ Nn.

Note that when R is a polynomial ring, the above definition is consistent with thegeneral definition of Grobner basis in polynomial rings.

In this paper, a signature-based algorithm for computing Grobner basis in thequasi-commutative ring R is proposed. This new algorithm has a similar structurewith the F5 and GVW algorithms for computing Grobner basis in polynomial rings,but it is not only a simple copy of F5 or GVW, because R is not commutative.Complete proofs for the correctness and termination of this algorithm are alsopresented in this paper.

27

Computing Real Solutions of Polynomial Systems viaLow-rank Matrix Completion

ê�!|wùKey Laboratory of Mathematics Mechanization, AMSS

{yma, lzhi}@mmrc.iss.ac.cn

Polynomial system solving is a classical and well studied problem in mathe-matics with practical applications in many areas. Various relevant questions canbe reformulated into the task of finding common real roots of a given system ofpolynomials in several variables:

p1(x1, . . . , xn) = 0p2(x1, . . . , xn) = 0

...ps(x1, . . . , xn) = 0

where pi ∈ R[x1, . . . , xn] for all i = 1, . . . ,m.In this paper we propose a novel algorithm for computing real roots of a system

of polynomial equations or real roots satisfying prescribed polynomial inequalitiesps+1 ≥ 0, . . . , pm ≥ 0. We apply low-rank matrix completion method in [4] to solvea hierarchy of LMI relaxations:

min ||Mt(y)||∗s. t. y0 = 1

Mt(y) � 0Mt−dj (pj y) = 0 j = 1, . . . , sMt−dj (pj y) � 0 j = s+ 1, . . . ,m

where dj := ddeg(hj/2)e, j = 1, . . . ,m. If at some order t one can find an optimalsolution y∗ to the above moment relaxation which satisfies the rank condition

rank Mt(y∗) = rank Mt−d(y

∗)

where d = max1≤j≤mdj , then one can extract real roots for the original polynomialsystem, see [1, 2, 3, 5].

There’s no guarantee for our algorithm to compute all real roots of a systemof polynomial equations p1 = 0, . . . , ps = 0. However, if there is only one root ofa large scale polynomial system in a given area defined by a semialgebraic set, wecan extract it quickly. Moreover, our algorithm can also be used to find some realsolutions on manifolds.

References

[1] G. Chesi, A. Garulli, A. Tesi, and A. Vicino. Characterizing the solution setof polynomial systems in terms of homogeneous forms: an LMI approach. Int.Journal of Robust and Nonlinear Control, 13(13):1239–1257, 2003.

28

[2] Didier Henrion and Jean Bernard Lasserre. Detecting global optimality andextracting solutions in GloptiPoly. In D. Henrion and A. Garulli, editors, Posi-tive Polynomials in Control, volume 312, pages 293–311. Springer Verlag, 2005.Lecture Notes on Control and Information Sciences.

[3] Jean B. Lasserre, Monique Laurent, and Philipp Rostalski. Semidefinite charac-terization and computation of zero-dimensional real radical ideals. Foundationsof Computational Mathematics, 8(5):607–647, 2008.

[4] Yue Ma and Lihong Zhi. The minimum-rank Gram matrix completion viamodified fixed point continuation method. In ISSAC 2011: Proceedings of the36th international symposium on Symbolic and algebraic computation, pages241–248. ACM, 2011.

[5] Markus Schweighofer. Optimization of polynomials on compact semialgebraicsets. SIAM Journal on Optimization, 15(3):805–825, 2005.

The N−fold Darboux transformations and exact solutions of theBoussinesq-Burgers equation

rï� �CSchool of mathematical sciences,Dalian University of

Technology,Dalian,116024,Chinameijq@dlut.edu.cn mengy926@163.com

A systematic method to construct N-fold Darboux transformations of Bousinesq-Burgers equation has been presented. By using symbolic computation, two typesof N-fold Darboux transformations have been obtained. And according to the Dar-boux transformations, some new exact solutions of the equation are obtained.

KeywordµBoussinesq-Burgers equation, Darboux transformation, Exact solu-tion, symbolic computation

Computing symbolic determinants by approximate interpolation

���1,2, ¾]1, �²�1,2, ܵ¥1

1Laboratory for Automated Reasoning and Programming, Chengdu Institute ofComputer Applications, CAS, Chengdu 610041, China

2Graduate School of the Chinese Academy of Sciences, Beijing 100049, China{qinxl,yongfeng}@casit.ac.cn

An effective algorithm is presented for computing exact determinants of ma-trices with polynomial entries by approximate interpolation. Such a problem isimportant in many applications. The algorithm relies on error analysis of approx-imate Newton’s interpolation method for solving Vandermonde systems. It is also

29

based on a novel approach for estimating the degree of variables, and the Kroneckertrick algorithm for dimension reduction. From this approach, the parallelization ofthe method arises naturally.Keyword: symbolic determinants, approximate interpolation, symbolic-numericalcomputation, Vandermonde systems

���������555ëëëYYYXXXÚÚÚ­­­½½½555���###���âââ

d�§û¡J>f�E�Æ, oA 611731

JÑ��u����5ëYXÚ­½5�#�{§T�{Äuõ�ª�OXÚnا�±éXÚ­½�Ä�Ñ����ä§�ÑDÚª��{I���þªÇ:?1u���UJøCq(Ø�Øv"�­��´§d�{�±^u©Û¹ëêXÚ�­½üÝ"¢S~fw«T�{�k�5"

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Camera Calibration fromthe Rotation Matrix Parameterized by Unit Quaternion

)ïû He-Zhi Wang Wen-Peng Ma

College of Mathematics and Computational ScienceShenzhen University, Shenzhen, Guangdong 518060, P. R. China

E-mail: jtang@szu.edu.cn, wanghz020111@126.com,markwinpe@163.com

The camera calibration is a fundamental problem in computer vision, the studyof it has made great development in recent years. A new camera calibration tech-nique which can be finished by using single image of objects combined by two 1-Dobjects is introduced in “Simple camera calibration from a single image using fivepoints on two orthogonal 1-D objects”, it is an effective camera calibration methodas a result of using single image of object. But the rotation matrix is parameterizedby Euler angles, the value of sine and cosine function will be changed a lot when theradian of an angle has a small fluctuation. Moreover, gimbal lock appears some-times adding difficulties in solving angles. In order to avoid these problems, we useUnit Quaternion instead of Euler angles to parameterize the rotation matrix.

We assume that unit quaternion q1 = a1 + (0, 0, d1), q2 = a2 + (b2, 0, 0), q3 =a3 + (0, 0,−d3) express rotation around the Z−, X−, Zw− axes by the angles θ, φand ω respectively, then R1, R2, R3 are rotation matrixes produced by q1, q2, q3, sothe matrix R transforming vectors from the world coordinate system: XwYwZw tothe camera coordinate system: XY Z is expressed as R = R1R2R3. The reference

30

object that can be used in our camera calibration method consists of two orthogonal1-D objects: lineL1 and lineL2, which cross at point O, points A and B are set onlineL1, points C and D are set on lineL2. We assume that line L1 is in agreementwith the Zw − axis and line L2 is in agreement with the Yw − axis, point O isthe origin of the world coordinate system. We rotate the camera and L2 aroundZw − axis by ω, then the camera’s rotation matrix is R = R1R2. We also assumethat the aspect ratio is one, skew is zero, principal point is projection center, andthere is no lens distortion, f is the focal length. According to camera model, wecan express the coordinate of image points of O, A, B, C and D by the parameterof the camera, then q1,q2,q3 will be found when the range of rotation angle is sure.The 3-D space is divided eight octants by the world coordinate system and therange of angles θ, φ and ω in different octant is determined uniquely, so we canget q1, q2 and q3 when the camera in different octant. Finally, we obtain cameraposition (Tx, Ty, Tz)T by solving simultaneous linear equations.

Simulative experiments are carried out to show validity and practicability ofthis method.

(This work is supported by the National Natural Science Foundation of China(No.61075037) and Shenzhen Municipal Science and Technology Plan(JC200903130224A,JC201005280522A).)

Determination and Parametrization ofrational developable surfaces (Ongoing)

�á] Ç=ÀSchool of Mathematical Sciences, Graduate University of CAS, Beijing 100049,

Chinalyshen@gucas.ac.cn

The rational developable surfaces are widely used in industrial manufactory.Generally, they are defined in parametric form and designed from CAGD and CAD.On the other hand, a surface can also be given in algebraic form may be developableone. Then for a given algebraic surface, we try to determine that whether it is de-velopable or not. As known that the parametric representations of the surfaces aremore suitable in model design and drawing graphics, we will parameterize the alge-braic surfaces if they are developable. In parameterization of the rational algebraicdevelopable surfaces, the the directrices are firstly found and parameterized, thenthe different developable surfaces can be parameterized by computing the rulingdirections, vertices and crest curves respectively.

It is proved that a differentiable surface is developable if and only if its Guassiancurvature equals zero. Then the first work is to compute the Guassian curvature of agiven algebraic surface. The Guassian curvature of an algebraic surface is based onthe theorem of implicit function. For the next consideration, if the algebraic surfaceis developable, it is necessary to determine whether it can be rational parameterizedor not.

31

The rationality of a given algebraic surface is determined by its genus. Forthe case of a developable surface, the second plurigenus is always zero. For thearithmetic genus, it is known that it is equal to the genus of the crest curve, theone of which the surface is the tangent developable of. So a developable surface isrational if and only its crest curve is rational. An exception is the cone, which isdevelopable but has no crest curve; this is rational if and only if the base curve (i.e.directrix) is rational. The cylinder is also a cone with its apex at the infinite.

For a given developable algebraic surface, we can determine it as a cylinder, acone or a tangent surface. For the cylinder and cone surface, we need to computethe base curves. For the tangent surface, one need to figure out its crest curve.If these curves are rational, then find they have rational parametrizations. Basedon the parametrizations of these characteristic curve, a rational of the developablesurface can be computed. The main process is illustrated in the following algorithm.

Algorithm 0.3 Input: An irreducible algebraic surface F (x, y, z) = 0.Output: A rational parametrization of F (x, y, z) = 0 if exists.

1) If the surface F (x, y, z) = 0 is developable, goto Step 2). Otherwise, thealgorithm terminates.

2) if F = 0 is a rational cylinder, find a rational parametrization P (s) of a basecurve C = 0. Output (P (s)) + (x0, y0, z0)t, where (x0, y0, z0) is the directionof the rulings.

3) if F = 0 is a rational cone, find a rational parametrization P (s) of a basecurve C = 0. Output (x0, y0, z0) + P (s)t, where (x0, y0, z0) is the vertex ofF (x, y, z) = 0.

4) if F = 0 is a rational tangent surface, find a rational parametrization P (s)of the crest curve of F = 0. Output P (s) + P ′(s)t.

In the above process, there are several subalgorithms needed such as determin-ing the developablity, classifying the type of the developable surface, computing thebase curve and the crest curve, and rationally parameterizing these curves.

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�Ä��C�Îé��m�Ú2m�ûI£�¹�ÎûI��¤5O�u¢�(Ç,ùpm´zaûI¥�ûIê£�©m=10¤"·���{Úã�/G£O¥²;��{–Fourier£ãf±9ZernikeÝ�'�"¢�(JXL1§§L²ÃØ´First-tierÚSecond-tier§�©JÑ�V-£ãf�{Ñ`uFourier£ãfÚZernikeÝ"��~f§ã1�ÑÙ¥üÌûI��[u¢ÑÑ(J§3ã1�z��㥧�ý��ã´�ÎûI§mý20��ã´u¢��c20�(J§U�q§Ýl���ü�"ûI��q5u¢ØÓu���/G��§3ûIã��Óaã�¥§��Ø

�q�ã�Øõ§�q��Ué�§���Ó�A�vk"ûIã��u¢Ò´�çÀÑ��Îã��qÝ'�p�ã�"�©æ^��VXÚéûI?1u¢§duV-XÚU°(L�z��ûIã��Ó+§AO´�ûIã�´dõ�©l�Ü©|¤�£ù3ûIã�¥�þÑy¤§�wÑV-XÚ°(L��`³§Ïd§3J�ûI�A��þ�k�p�O(5Ú�&ݧg,3ûIu¢�¡ULyÑ`³"

34

Figure 1: *

Figure 2: *(a)V-£ãf

Figure 3: *(b)Fourier£ãf

Figure 4: *(c)ZernikeÝ

Figure 5: *(g)V-£ãf

Figure 6: *(h)Fourier£ãf

Figure 7: *(i)ZernikeÝ

Figure 8: *ã1 u¢ÑÑ(J'�Þ~

35

An Efficient Method ofComputing Comprehensive Grobner Bases and

Minimal Comprehensive Grobner Bases

��, �½xKLMM, Academy of Mathematics and Systems Science, CAS, Beijing 100190,

Chinasunyao@amss.ac.cn, dwang@mmrc.iss.ac.cn

A new and efficient approach is proposed for computing a comprehensive Grobnerbasis (CGB) and minimal comprehensive Grobner basis (miniCGB) of a parame-terized polynomial system. This new technique fully uses existing algorithms forcomputing comprehensive Grobner systems (CGS). CGB and CGS are defined byWeispfenning, and they can be used to solve many engineering problems which in-volve parameters. Another important application of CGB and CGS is automaticgeometric theorem proving.

The definitions of CGB and CGS are presented as follows. Let k be a field, R bethe polynomial ring k[u] in the parameters u = {u1, · · · , um}, and R[x1, · · · , xn]be the polynomial ring over the parameter ring R in the variables x1, · · · , xn ={x1, · · · , xn} and x1, · · · , xn ∩u = ∅. Let L an algebraically closed field containingk. A specialization of R is a homomorphism σ : R −→ L and the specialization σextends canonically to a specialization σ : R[X] −→ L[X] by applying σ coefficient-wise. A specialization σa deduced by a point a ∈ Lm is defined as: σa : f −→ f(a)where f ∈ R.

Definition 8 (CGB) Let F be a subset of R[X] and S be a subset of Lm. Afinite subset G ⊂ 〈F 〉 ⊂ R[X] is called a comprehensive Grobner basis on S for F ,if σa(G) is a Grobner basis of the ideal 〈σa(F )〉 ⊂ L[X] for a ∈ S.

A comprehensive Grobner basis G on S for F is said to be minimal, if theredoes not exist a proper subset G′ ( G such that G′ is also a comprehensive Grobnerbasis on S for F .

Definition 9 (CGS) Let F be a subset of R[X], A1, · · · , Al be algebraically con-structible subsets of Lm, G1, · · · , Gl be subsets of R[X], and S be a subset of Lm

such that S ⊆ A1 ∪ · · · ∪ Al. A finite set G = {(A1, G1), · · · , (Al, Gl)} is called acomprehensive Grobner system on S for F , if σa(Gi) is a Grobner basis of the ideal〈σa(F )〉 ⊂ L[X] for a ∈ Ai and i = 1, · · · , l.

The new method for computing comprehensive Grobner bases can be appliedupon almost all existing algorithms for computing comprehensive Grobner systems.The key idea is not to simplify a polynomial under various specialization of itsparameters, but rather keep track in the polynomial, of the power products whosecoefficients vanish; this is achieved by partitioning the polynomial into two parts–nonzero part and zero part for the specialization under consideration. During the

36

computation of a comprehensive Grobner system, for a particular branch corre-sponding to a specialization of parameter values, nonzero parts of the polynomialsdictate the computation, i.e., computing S-polynomials as well as for simplifyinga polynomial with respect to other polynomials; but the manipulations on thewhole polynomials (including their zero parts) are also performed. Grobner basiscomputations on such pairs of polynomials can also be viewed as Grobner basiscomputations on a module. Once a comprehensive Grobner system is generated,both nonzero and zero parts of the polynomials are collected from every branchand the result is just a comprehensive Grobner basis. By doing extra operations onthese nonzero and zero parts of the polynomials, a minimal comprehensive Grobnerbasis can be constructed as well..

The Optimal Linear Secret Sharing Scheme for Any Given AccessStructure

/S²1, pXö, ܤw1Guangzhou University, Guangzhou, 510006 China

ctang@gzhu.edu.cn

Any linear code can be used to construct a linear secret sharing scheme. Inthis paper, it is shown how to decide optimal linear codes (i.e., with the biggestinformation rate) realizing a given access structure over finite fields. It amounts tosolving a system of quadratic equations constructed from the given access structureand the corresponding adversary structure. The system becomes a linear systemfor binary codes. An algorithm is also given for finding the adversary structure forany given access structure.

Keyword: Cryptography, secret sharing, linear code, access structure, adversarystructure.

Solution Classification forPerspective-Three-Point Problem based on PST Method

)ïû Peijian Wang

College of Mathematics and Computational ScienceShenzhen University, Shenzhen, Guangdong 518060, P. R. China

E-mail: jtang@szu.edu.cn, wangpeijian2008@126.com

The Perspective-n-Point(PnP) problem is originated from camera calibration.Also known as pose estimation, it is to determine the position and orientation of thecamera with respect to a scene object from n correspondent points. The P3P prob-lem is the smallest subset of control points that yields a finite number of solutions.

37

What’s more, all other PnP (n > 3) problems include the P3P problem as a specialcase. Therefore, a study of this problem is desirable. Gao etc.[1] used Wu-Ritt’szero decomposition algorithm to give a complete triangular decomposition for theP3P equation system. This decomposition gives a complete solution classificationfor the P3P equation system. Li and Xu [2] presented a new direct solution of P3Pproblem with high numerical stability and accuracy. The main idea is to reducethe number of variables by using a geometric constraint called ”Perspective Simi-lar Triangle”(PST). And they given a new equation system about P3P problem asfollows: t22 −A1t

21 +A2 = 0

A3t21 +A4t1t2 +A5t1 +A6t2 +A7 = 0

(1)

where Ai(i = 1, ..., 7) are known as parameters, tj(j = 1, 2) are variables.In this paper, we convert the new P3P problem equation system (1) to (2), as

follows: B1t

41 +B2t

31 +B3t

21 +B4t1 +B5 = 0

t2 =√A1t21 −A2

(2)

where Bi(i = 1, ..., 5) are known as parameters. And then, using the completediscrimination system [3], we give the solution classification of t1 from the firstequation of (2). The solution classification gives a set of formulas to determine thenumber of real solutions to the P3P problem. Based on the formulas, we may knowwhether the parameters give multiple solutions or not and are critical or not whichis very important to present robust algorithm.

References

[1] Xiaoshan Gao and Xiaorong Hou etc., Complete Solution Classification for thePerspective-Three-point Problem. IEEE Transactions on Pattern Analysis andMachine Intelligence, VOL.25, NO.8, AUGUST 2003, 930-943.

[2] Shiqi LI and Chi Xu, A Stable Direct Solution of Perspective-Three-PointProblem. International Journal of Pattern Recognition and Artificial Intelli-gence. IJPRAI-D-10000178.

[3] L.Yang, J.Z.Zhang, and X.R.Hou, Non-linear Equation System and Auto-mated Theorem Proving, Shanghai Press of Science Technology and Educa-tion,1996.(in Chinese)

(This work is supported by the National Natural Science Foundation of China(No.61075037) and Shenzhen Municipal Science and Technology Plan(JC200903130224A,JC201005280522A).)

38

���þþþ������>>>kkknnn������¯KKK���Fitzpatrick-Neville...���{{{

g*, Üäõ, XA3��ÆêÆïĤ

xpxiapengxp@163.com, lein@jlu.edu.cn

Neville�{´dEric Harold NevilleJÑ�. T�{�8�´��O���õ�ª3�½:?���([3]). Stoer3©z[3]¥�Ñ��kn��¯K�Neville.�{. Sweatman[4] ò��kn���Neville.�{í2��þ�/, ïá���þ�kn��¯K�Bulirsch-Stoer-Neville.�{. ¢Ã�´ù�{ÑJ±í2�õ��/. �(�!ô²[5]±ë©ª�Ä:�Ñ(��Úõ�)kn���Neville.�{. ^T�{O����, e����û�ÓIN�!:^S­#O�.�©òFitzpatrick�{[1, 2]A^��þ��>kn��¥, ¿|^Hermite �

�ļê�5�, éFitzpatrick �{?1·�?�, |^T?��Fitzpatrick�{,�ØO�f��é(~a, b)�äNL�ª, ��O�Ñõ��þ��>kn��¯K�f��é(~a, b)3�½:Y0?��, ? O��þ��>kn��¼ê3�½:���, ¡?��Fitzpatrick�{�Fitzpatrick-Neville.�{. T�{�`:´�±¯�O��þ��>kn��¼ê3�½:���.�þ��>kn��¯Kµ�½L�ØÓ�:¤�¤�8Ü{Yi ∈ Rn : i =

1, . . . , YL}, �A�k���þ8{~V α(Yi) = ~V αi = (V α

i,1, . . ., Vαi,d) ∈ Rd : i =

1, . . . , L}, Ù¥α ∈ Ai, Ai (i = 1, . . . , L)�lower set, Ïé�þ�kn©ª¼ê

~r(X) =~a(X)

b(X). (5)

¦Ù÷v

Dα~r(Yi) = Dα~a(Yi)

b(Yi)= ~V α

i , α ∈ Ai, i = 1, · · · , L, (6)

Ù¥ ~a(X) ´ d ��þõ�ª, = ~a(X) = (a1(X), · · · , ad(X)), aj(X) (1 ≤ j ≤d) ´R[X] = R[x1, . . . , xn]¥�õ�ª, b(X) ´õ�ª, �q(Yi) 6= 0, ~r(X) =

(r1(X), · · · , rd(X)), Dα =1

α1! · · ·αn!

∂α1+···+αn

∂xα11 · · · ∂x

αnn��©�f.

(6)ª��d/ª�~a(Yi)

b(Yi)3Yi:�Taylor�

~a(Yi)

b(Yi)=∑α∈Ai

~V αi (X − Yi)α +

∑α/∈Ai

Dα~a(Yi)

b(Yi)(X − Yi)α. (7)

Psi = ]Ai, i = 1, . . . , L, N =L∑i=1

si. òAi¥���­ü§¦�Ai�z�f

8Ai,j = {αi,0, . . ., αi,j}, 0 ≤ j ≤ si − 1, E,´lower set. AO�Ai,si−1 = Ai.

39

-Ii,j�(Yi,Ai,j)¤(½��²n�, =Ii,j = I((Yi,Ai,j)) = {p ∈ P : Dαp(Yi) =

0, ∀α ∈ Ai,j}. ½Â~hi =∑

α∈Ai

~V αi (X − Yi)α.

½½½ÂÂÂ1¡(~a(X), b(X))��þ��>kn��¯K�f��é,XJ(~a(X), b(X))÷v

~a(X) ≡ b(X)~hi mod Ii, i = 1, . . . , L, (8)

Ù¥Ii�d:YiÚlower set Ai¤(½��²n�.PP = R[X]. é∀(~a1, b1), (~a2, b2) ∈ Pd+1, ∀c ∈ P, ½Â(~a1, b1) + (~a2, b2) =

(~a1 + ~a2, b1 + b2), c(~a, b) = (c~a, cb), Kd�þ��>kn��¯K�¤kf��

é�¤�8ÜM = {(~a, b) : ~a ≡ b~hi mod Ii,si−1, i = 1, . . . , L}�Pd+1�f�. b�{(~a1, b1), . . . , (~amN

, bmN)}�M�GrobnerÄ, K?Ûf��é(~a, b)�L«�

(~a, b) = c1(~a1, b1) + · · ·+ cmN(~amN

, bmN), ci ∈ P, i = 1, . . . ,mN .

ÀJ·��ëêci¦�(c1b1 + · · · + cmNbmN

)(Yi) 6= 0, i = 1, . . . , L, K(~a, b)��þ��>kn��¯K).|^Fitzpatrick�{O��þ�kn��¯K§I½ÂXe�ü�Sµ½½½ÂÂÂ2(Pd+1þ�ü�S≺ ∂

∂xi

)

1. ¡Xβ(~ej , 0) ≺ ∂∂xi

Xα(~ei, 0) e|β| < |α|, ½|α| = |β|�j < i, ½|α| = |β|,i = j �Xβ ≺lex Xα,

2. ¡Xβ(~0, 1) ≺ ∂∂xi

Xα(~ei, 0) e|β|+ ∂∂xi

< |α|,3. Xβ(~0, 1) ≺ ∂

∂xi

Xα(~0, 1) e|β| < |α| ½|α| = |β| �Xβ ≺lex Xα,

Ù¥≺lex �P¥�i;S, ∂∂xi���½�ê.

�±�y≺ ∂∂xi

�Pd+1þ�ü�S.½Âmax{tdeg(~a), tdeg(b)+ ∂∂xi}�(~a, b)�E

,Ý(complexity) Ù¥tdeg(~a) = max{tdeg(ai) : i = 1, . . . , d}. ü�S≺ ∂∂xi

��^

´^5��f��é(~a, b)�E,Ý.

éz�Ai,j , 1 ≤ i ≤ L, 0 ≤ j ≤ si − 1, ½ÂÓ{�§~a ≡ b~hi mod Ii,j , Ù¥Ii,j�(Yi,Ai,j)¤(½��²n�.élower sets {Ai,j , 1 ≤ i ≤ L, 0 ≤ j ≤ si− 1}½ÂS<, ¦Ù÷vµAi,j1 < Ai,j2

��=�Ai,j1 ⊂ Ai,j2 , =j1 < j2. �i1 6= i2�Ai1,j1ÚAi2,j2�m�üS�±?¿ÀJ, �Qã�B�i1 6= i2�, ½ÂAi1,j1 < Ai2,j2��=�i1 < i2. éA�, �½Â

Ó{�§�S: ~a ≡ b~hi mod Ii1,j1 < ~a ≡ b~hi mod Ii2,j2 ��=�Ai1,j1 < Ai2,j2 .l ïákÚ(i, j)�m���N�:

k ←→ (ik, jk)

½Âf�S�Mk, k = 0, . . . , N , Ù¥M0 = Pd+1, Mk�ck�Ó{�§�)¤�¤��, =

Mk = {(~a, b) ∈Mk−1 : ~a ≡ b~hik mod Iik,jk , }, k = 1, . . . , N. (9)

M = MN = {(~a, b) : ~a ≡ b~hi mod Ii,si−1, i = 1, . . . , L}, ¿�M0 ⊃M1 ⊃ · · · ⊃MN . w,〈(~e1, 0), . . ., (~ed, 0), (~0, 1)〉 �M0�GobnerÄ. �½Pd+1ü�S≺ ∂

∂xi

, |

^Fitzpatrick�{�4í/O�M�GrobnerÄ.

40

Fitzpatrick algorithm VVRI :Input: Mk�4�GrobnerÄGk = {(~a1, b1), . . . , (~amk

, bmk)};

Output: Mk+1�4�GrobnerÄGk+1;Gk,0 := Gk;for t from 1 to d do

mk,t−1 := ]Gk,t−1;­üGk,t−1¥���¦Ù÷v

Gk,t−1 = {(~a1, b1), . . . , (~amk,t−1, bmk,t−1

)}¿�LT(~a1, b1) ≺ ∂∂xi

· · · ≺ ∂∂xi

LT(~amk,t−1, bmk,t−1

);for s from 1 to mk,t−1 do

bshik+1,t − as,t ≡ νs,t(X − Yik+1)αik+1

,jk+1 mod Iik+1,jk+1; (10)

end do;é���sk,t�¦�νsk,t,t 6= 0; (eé¤k�sþkνs,t = 0KGk,t :=

Gk,t−1;)for j from sk,t + 1 to mk,t−1 do

(~as, bs) := (~as, bs)−νs,t

νsk,t,t(~ask,t

, bsk,t);

end do;

Gk,t :={

(~a1, b1), . . . , (~ask,t−1, bsk,t−1),(~ask,t, bsk,t

) · (x1 − yik+1,1), . . . ,

(~ask,t, bsk,t

)·(xn−yik+1,n),(~ask,t+1, bsk,t+1), . . .,(~amk,t−1, bmk,t−1

)}

;

Gk,t :=minimal Grobner basis(Gk,t);end do;return Gk+1 = Gk,d;3Fitzpatrick algorithm VVRI�{¥, ·�òbshik+1,t − as,t ( mod I)L«

¤Hermite��ļê�5|Ü,|^Hermite��ļê�5�(ÃIO�Hermite��ļê�L�ª)�4í/O�Ñνs,t. ?�ÚòO�(~as, bs)�L�ªU¤O�(~as, bs)3Y0?��, l 4í/O�ÑM�GrobnerÄ

{(~a1, b1), . . . , (~amN, bmN

)}

3Y0:?��, =®�{(~a1(Y0), b1(Y0)), . . ., (~amN(Y0), bmN

(Y0))}, ?�ci ∈ R, ¦Ù÷vc1b1(Y0) + · · ·+ cmN

bmN(Y0) 6= 0, ����¼ê3Y0?���

c1~a1(Y0) + · · ·+ cmN~amN

(Y0)

c1b1(Y0) + · · ·+ cmNbmN

(Y0),

�©�ci = sgn(bi(Y0))(bi(Y0)�ÎÒ).

References

[1] O’Keeffe, H., Fitzpatrick, P., 2000. Recursive construction of groebner bases forthe solution of polynomial congruences. In: Brian Marcus, J. R. (Ed.), Codes,

41

Systems and Graphical Models. IMA, Volumes in Mathematics and its Applica-tions. Springer-Verlag, New York, pp. 299–311.

[2] O’Keeffe, H., Fitzpatrick, P., 2002. Groebner basis solutions of constrainedinterpolation problems. Linear Algebra and its Applications 351–352, 533–551.

[3] Stoer, J., Bulirsch, R., 1993. Introduction to Numerical Analsis, 2nd Edition.Springer, New York.

[4] Sweatman, W. L., 1998. Vector rational interpolation algorithms of bulirsch-stoer-neville form. Proceedings: Mathematical, Physical and Engineering Sci-ences 454 (1975), 1923–1932.

[5] Tan, J. Q. Jiang, P., 2004. A Neville-like method via continued fractions, Journalof Computational and Applied Mathematics 163, 219-232

A Majorization Order onMonomials and Termination of

the Successive Difference Substitution Algorithm

MWCollege of Computer Science and Technology, Southwest University for

Nationalities, Chengdu, Sichuan 610041, PR China E-mail: j.jia.xu@gmail.com

�]Chengdu Institute of Computer Applications, Chinese Academy of Sciences,

Chengdu,Sichuan 610041, PR China E-mail: yaoyong@casit.ac.cn

Abstract When the ordering of variables is fixed (e.g., x1 ≥ x2 ≥ · · · ≥ xn), the

monomial Xα = xα11 · · ·xαn

n majorizing the monomial Xβ = xβ1

1 · · ·xβnn (|α| = |β|)

means that∑ki=1 αi ≥

∑ki=1 βi (k = 1, · · · , n−1). SDS is a algorithm that can solve

harder problems with lesser mathematics and NEWTSDS is a improved algorithmof SDS. They are all special cases of a general successive difference substitutionalgorithm called as KSDS. In this paper, a necessary condition of positively ter-minating of KSDS for an input f is obtained by using a majorization order onmonomials. That is, every single term with negative coefficients in the form f ismajorized at least by a single term with positive coefficients of f in an arbitraryordering of variables.

Keywords Successive difference substitution algorithm, majorization order onmonomials, termination, positive semi-definite formMR(2000) Subject Classification 68T15 26D05

(The work of the authors were supported by the National Natural Science Foun-dation of China(11001228).The work of the second author was supported by the

42

the Open Project of Shanghai Key Laboratory of Trustworthy Computing of Chinaand NNSFC 91018012.)

Analytical nonautonomous matter-wave solutions inBose-Einstein condensate

A�æKLMM, Institute of Systems Science, Chinese Academy of Sciences China

zyyan@mmrc.iss.ac.cn

In this talk, we report a physical model that describes the transport of Bose-Einstein-condensed atoms from a reservoir to a waveguide. By using the similar-ity and M?bius transformations, we study nonautonomous matter waves in Bose-Einstein condensates in the presence of an inhomogeneous source. As a conse-quence, we obtain its various types of exact nonautonomous matter-wave solutions,including the W-shaped bright solitary waves, W-shaped and U-shaped dark soli-tary waves, periodic wave solutions, and rational solitary waves. These results showthat these different types of matter-wave structures can be generated and effectivelycontrolled by modulating the amplitude of the source.

Matrix formulae of differential resultant fora class of generic ordinary algebraic differential polynomials

Ü�]!�S²!p�ìKLMM, Institute of Systems Science, AMSS, Chinese Academy of Sciences,

Beijing 100190, P.R. Chinazhiyong-2008@163.com

Multivariate resultants for differential polynomials were defined using an idealapproach by Gao, Li and Yuan in [1]. It is well known to find explicit representationfor resultant is one of the major issues in studying resultant, and whether themultivariate differential resultant admits a matric representation is a major openproblem for differential resultant. In this paper, based on the idea of Macaulayalgebraic resultants, we present the matrix formulae of the differential resultant fortwo generic ordinary algebraic differential polynomials with order one and arbitrarydegrees. The constructed matrix is nonsingular and is used to show some propertiesof differential resultant directly. Although very special, this is the first matricrepresentation for a class of non-linear differential polynomials.

References

[1] X.S. Gao, W. Li, C.M. Yuan. Intersection Theory in Differential AlgebraicGeometry: Generic Intersections and the Differential Chow Form. Acceptedby Trans. Amer. Math. Soc., 58 pages. Also in arXiv:1009.0148v1.

43

Semi-algebraically Connected Components ofMinimum Points of a Polynomial Function

Q2,§Xiao ShuiJingDepartment of Mathematics, Nanchang University, Nanchang 330031, China

E-mail: zenggx@ncu.edu.cn, xiaoshjing@163.com

Recently, we provided an effective algorithm for catching possible points relatedto the global infimum and minimum of a given multivariate polynomial, see Algo-rithm 4.3 in reference [7]. Based on this algorithm, we obtained another algorithmfor both computing the global infimum and deciding its attainability.

The present paper is a complement of reference [7]. In this paper, it is fur-ther shown that Algorithm 4.3 in [7] is able to find at least one point in eachsemi-algebraically connected component of minimum points for a given multivari-ate polynomial having its global minimum. Explicitly speaking, such a result isobtained as follows:

Let f ∈ F [x1, ..., xn] be a non-zero polynomial, where F is a com-putable ordered subfield of R, and assume that the global infimum off is attained. If a family Ξ of Rational Univariate Representationsis obtained by Algorithm 4.3 in [7], then, for every semi-algebraicallyconnected component D of minimum points of f , (0, ..., 0) ∈ D or Dcontains a point which may be created by standardizing some ∂

∂xi∈ Ξ.

In order to create at least one minimum point in every connected component bystandardizing some ∂

∂xi∈ Ξ, an algorithm is also provided in this paper. Finally,

this paper gives two examples to illustrate our algorithms

References

[1] Hagglof K, Lindberg P O, Stevenson L. Computing global minima to polyno-

mial optimization problems using Grobner bases. J Global Optimization, 1995,

7: 115–125

[2] Uteshev A Y, Cherkasov T M. The search for the maximum of a polynomial.

J Symbolic Comput, 1998, 25: 587–618

[3] Parrilo P A, Sturmfels B. Minimizing polynomial functions. In: Algorithmic

and quantitative real algebraic geometry, DIMACS Series in Discrete Math-

ematics and Theoretical Computer Science 60. Providence: Amer Math Soc,

2003, 83–100

44

[4] Hanzon B, Jibetean D. Global minimization of a multivariate polynomial using

matrix methods, J. Global Optimization, 2003, 27: 1–23

[5] Nie J, Demmel J, Sturmfels B. Minimizing polynomials via sums of squares

over the gradient ideal, Math Program Ser A, 2006, 106: 587–606

[6] Guo F, Safey EI Din M, Zhi L H. Global Optimization of Polynomials Us-

ing Generalized Critical Values and Sums of Squares. ISSAC 2010, Munich,

Germany.

[7] Xiao S J, Zeng G X. Algorithms for computing the global infimum and mini-

mum of a polynomial function (in Chinese). Sci Sin Math, 2011, 41(9): 759–788

[8] Zeng G X, Xiao S J. Computing the rational univariate representations for

zero-dimensional systems by Wu’s method (in Chinese). Sci Sin Math, 2010,

40(10): 999–1016

[9] Bochnak J, Coste M, Roy M F. Real Algebraic Geometry. New York-Berlin-

Heidelberg: Springer-Verlag, 1998

[10] Basu S, Pollack R, Roy M F. Algorithms in Real Algebraic Geometry, Algo-

rithms and Computation in Math. 10. Berlin: Springer-Verlag, 2003

[11] Wu W T. Mathematics Mechanization: Mechanical Geometry Theorem-

Proving, Mechanical Geometry Problem-Solving and Polynomial Equations-

Solving. Beijing/ Dordrecht-Boston-London: Science Press/Kluwer Academic

Publishers, 2000

[12] Zeng G X, Zeng X N. An effective decision method for semidefinite polynomi-

als. J Symbolic Comput, 2004, 37: 83–99

[13] Mishra B. Algorithmic Algebra, Texts and Monographs in Computer Science.

New York-Berlin-Heidelberg: Springer-Verlag, 1993

(This work is supported by the National Natural Science Foundation of China

(Grant No. 11161034).)

Some new upper bounds for Heilbronn Problems of

eight points in Squares and Triangles found by

numeric-symbolic computa

Q�Y, �û�

45

Shanghai Key Lab of Trustworthy Computing, East China Normal University,

Shanghai 200062

{zbzeng, lychen}@sei.ecnu.edu.cn

In this report we present our recent results in finding the following global opti-

mal problem: Let K be a square or a triangle, and P1, P2, . . . , Pn be points in the

set K. What is the maximal value ∆K(n) of the smallest area of the

(n3

)trian-

gles formed by the points? This is called Heilbronn Triangle Problem. Following is

the asymptotic estimation:

A · n−2 log n ≤ ∆(n) ≤ B · n−8/7eC√

logn,

where A,B and C are constants.

But the exact value of ∆K(n) is very difficult even for small integer n. The exact

results are known only for n ≤ 7 for squares and n ≤ 6 for triangles from literatures.

Lower bounds for n ≤ 16 have been found by D. Cantrell et al. Following are some

of the lower bounds.

∆S(8) ≥√

13−136 = 0.0723 · · · , ∆S(9) ≥ 9

√65−55320 = 0.0548 · · · ,

∆S(10) ≥ 0.046537 · · · ,

for squares, and

∆T (7) ≥ 772 , ∆T (8) ≥ 0.067789 · · · ,

∆T (9) ≥ 43784 = 0.0544847 · · · , ∆T (10) ≥ 0.043376 · · · ,

for triangles. But there not many works have been done for upper bounds. Actually,

a lower bound can be found through numerical search of local optimum, but the

upper bounds need complete search of global optimum. Recently, by using parallel

computation and a branch-and-bound-like approach, we have obtained the following

three results:

∆S(8) =

√13− 1

36, ∆T (7) =

7

72, ∆T (8) ≤ 0.067816.

The main sketch of our approach consists of the following three stages.

In the first stage, we work to find the a set of (pare-wise disjoint) very small

neighborhoods (squares or triangles) U1, U2, . . . , Un ⊂ K so that if {P1, P2, . . . , Pn}is an optimal configuration then

Pσ(1) ⊂ U1, Pσ(2) ⊂ U2, . . . , Pσ(n) ⊂ Un

46

for certain permutation {P1, P2, . . . , Pn} and certain congruent transformation of

K. We have used pure integer manipulation on a NUMA-Structure cluster com-

puter with 80 CPU cores and got square (and triangular) neighborhoods with di-

ameter less than 1/1024. The neighborhoods U1, U2, . . . , Un ⊂ K are obtained by

following sub-procedures.

Initial Construction: Divide the set K into 64 small congruent copies I1, I2, . . . , I64

so that I1⋃I2⋃· · ·⋃I64 = K, int(Ij)

⋂int(Ik) = φ, 1 ≤ j < k ≤ 64, and therefore

decompose Kn, the whole set of n-points configurations, into the family S0 of finite

set {I1, I2, . . . , I64}n. Then for each member Ij1 , Ij2 , . . . , Ijn in this family, estimate

the maximum of the following quantity

h(P1, P2, . . . , Pn) = min{area(PiPjPk), 1 ≤ i < j < k ≤ n}

over all P1, P2, · · · , Pn in Ij1 , Ij2 , . . . , Ijn . If this maximum is smaller than the

known lower bound of ∆K(n), then simply drop it, otherwise record this member

into S1 for further processing.

Recursive Processing: Let S1 be the set generated in the last step. For each

Ij1 , Ij2 , . . . , Ijn in S1, divide each Ijk into four small equal copies Ijk0, Ijk1, Ijk2, Ijk3,

so that Ij1 , Ij2 , . . . , Ijn is decomposed into 4n subsets in form Ij1l1 , Ij2l2 , . . . , Ijnln .

Then for all l1, l2, . . . , ln ∈ {0, 1, 2, 3} estimate the maximum of the quantity h(P1, P2,

. . . , Pn) over all P1, P2, . . . , Pn in Ij1l1 , Ij2l2 , . . . , Ijnln . If this value is equal or

greater than the known lower bound, record the subset Ij1l1 , Ij2l2 , . . . , Ijnln in the

set S2.

Recursively do the above processing, we will get a series S3, S4, S5, . . ., with each

Sk is constituted by a finitely many Ij1l1···m1 , Ij2l2···m2 , . . . , Ijnln···mn with 1 ≤j1, j2, · · · , jn ≤ 64, satisfying that diameter(Ijili···mi) = 1

2k+3 , where

k = #{ji, li, . . . ,mi} − 1,

and that if {P1, P2, . . . , Pn} is an optimal configuration, then it is contained in some

Ij1l1···m1 , Ij2l2···m2 , . . . , Ijnln···mn ∈ Sk for each k.

In the second stage, we figure out the shape of the optimal configurations

{P1, P2, . . . , Pn},

that is, to find the orientation of the triangle PiPjPk, and find the subset Λ of

{(i, j, k)|1 ≤ i < j < k ≤ n} which satisfies area(PiPjPk) > h(P1, P2, . . . , Pn) for

all (i, j, k) ∈ Λ. This is done by computing the following upper bound of ∆K(n):

UBk = max{max{h(P1, P2, . . . , Pn)|P1 ∈ Ij1l1···m1 , P2 ∈ Ij2l2···m2 , . . . , Pn ∈ Ijnln···mn},(Ij1l1···m1 , Ij2l2···m2 , . . . , Ijnln···mn) ∈ Sk},

47

and then comparing it with all {area(PiPjPk), 1 ≤ i < j < k ≤ n}.In the third stage, we transform the Heilbronn problems to non-linear optimizations

and devote to solve them. The non-linear optimizations are generated according to

the neighborhoods U1, U2, . . . , Un ⊂ K obtained in the first stage and the subset Λ

obtained in the second stage, then described by the following form:

max a,s.t. area(PiPjPk) > a, (i, j, k) ∈ Λ,

area(PiPjPk) ≥ a, (i, j, k) /∈ Λ,P i ∈ Ui, i = 1, 2, . . . , n.

Finally, we have solved the non-linear optimization problems corresponding to n = 8

for squares and n = 7 for triangles, which lead to the exact values of ∆S(8) =

(√

13 − 1)/36 and ∆T (7) = 7/72, with respectively. When K is a triangle and

n = 8, we were not able to solve the non-linear optimization completely, so we get

only an upper bound ∆T (8) ≤ 0.067816.

( Grants: National Natural Science Foundation of China (61021004,90718041),

Shanghai Municipal Natural Science Foundation (11ZR1411500), Innovation Pro-

gram of Shanghai Municipal Education Commission (11ZZ37).)

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On Arnold’s problem on the classifications of convex lattice polytopes

mD²

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In 1980, V.I. Arnold studied the classification problem for convex lattice poly-

gons of given area. Since then this problem and its analogues have been studied

by Barany, Pach, Vershik, Liu, Zong and others. Upper bounds for the numbers

of non-equivalent d-dimensional convex lattice polytopes of given volume or cardi-

nality have been achieved. Recently, by introducing and studying the unimodular

groups acting on convex lattice polytopes, we obtain lower bounds for the num-

ber of non-equivalent d-dimensional convex lattice polytopes of bounded volume or

given cardinality, which are essentially tight.

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50

On Convergence of the Inexact Rayleigh Quotient Iteration with

MINRES

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For the Hermitian inexact Rayleigh quotient iteration (RQI), we present a new

general theory, independent of iterative solvers for shifted inner linear systems. The

theory shows that the method converges at least quadratically under a new con-

dition, called the uniform positiveness condition, that may allow inner tolerance

bigger than one and can be considerably weaker than the existing condition in lit-

erature. We consider the convergence of the inexact RQI with the unpreconditioned

and tuned preconditioned MINRES method for the linear systems. Some attractive

properties are derived for the residuals obtained by MINRES. Based on them and

the new general theory, e make a more refined analysis and establish a number of

new convergence results, showing that the new convergence conditions are much

more relaxed than ever before. The theory can be used to design practical stopping

criteria to implement the method more effectively. umerical experiments confirm

our results.

Sparse Bivariate Polynomial Factorization

ǩ�

Motivated by Sasaki’s work in Hensel construction for solving multivariate al-

gebraic equations, we present a generalized Hensel lifting technique for bivariate

polynomial factorization over the rational field which takes the advantages of spar-

sity. Another feature is an efficient numeric combination method to solve extraneous

factor problem before the lifting stage. Meanwhile the leading coefficient problem

is also studied in this paper. Both theoretic analysis and experimental data show

that algorithm BiFactor presented in this paper is eficient, especially for sparse

polynomials and is applicable for most cases.

51

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A Greedy Algorithm for Feedrate Planning of

CNC Machines long Curved Tool Path with Confined Jerk

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The problem of time-optimal feedrate planning along a curved tool path for

3-axis CNC machines with acceleration and jerk bounds for each axis and the

tangential velocity bound is addressed. We prove that the optimal feedrate planning

must use “Bang-Bang” or“Bang-Bang-Singular” control, that is, at least one of the

axes reaches its acceleration or jerk bound, or the tangential velocity reaches its

bound throughout the motion. As a consequence, the time-optimal parametric

velocity can be expressed as a piecewise analytic function in the curve parameter u.

We also give the explicit formula for the velocity function when a jerk reaches its

bound by solving a second order differential equation. Under a “greedy rule”, an

algorithm for optimal jerk confined federate planning is presented, together with

two examples.

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53