第六期符号计算暑期讲习班
2019年7月2127日，中国重庆
Certified Numerical Real Root Isolation for Bivariate Polynomial Systems 程进三 中国科学院数学与系统科学研究院 摘要: In this paper, we present a new method for isolating real roots of a bivariate polynomial system. Our method is a subdivision method which is based on real root isolation of univariate polynomials and analyzing the local geometrical properties of the given system. We propose the concept of the orthogonal monotone system in a box and use it to determine the uniqueness and the existence of a simple real zero of the system in the box. We implement our method to isolate the real zeros of a given bivariate polynomial system. The experiments show the effectivity and efficiency of our method, especially for systems with high degrees and sparse terms. Our method also works for nonpolynomial systems. It is a joint work with Junyi Wen. 
Generalized Hermite Reduction, Creative Telescoping and Definite Integration of DFinite Functions Frédéric Chyzak INRIA, France 摘要: Hermite reduction is a classical algorithmic tool in symbolic integration. It is used to decompose a given rational function as a sum of a function with simple poles and the derivative of another rational function. In this talk, we extend Hermite reduction to arbitrary linear differential operators instead of the pure derivative, and develop efficient algorithms for this reduction. We then apply our generalized Hermite reduction to the computation of linear operators satisfied by single definite integrals of Dfinite functions of several continuous or discrete parameters. The resulting algorithm is a generalization of reductionbased methods for creative telescoping.

Computing the numerical rank of large matrices 李宗錂 国立中山大学 摘要: If rank deficiency for a large matrix is known to be small apriori, there seems no need to compute SVD to find all singular values for determining its rank. While the Householder QR with column pivoting algorithm proposed by Golub and Businger in 1965 works quite well in general for this purpose, there exist counter examples (by W. Kahan) that the method would fail. These failures of this sort had been overcome by T. Chan's RRQR algorithm in 1989. In practice, such as in signal processing, rank updatings and downdatings occur commonly along with the rankrevealing. While the UTV decomposition method for rank revealing proposed by G.W. Stewart in 1992 works quite well inrank updatings, it may not be efficient for the downdatings. Recently, an efficient algorithm for rank revealing is proposed. For the method, both updatings and downdatings become quite straightforward. The related results will also be presented in this talk. 
Factorization and Equivalence on Multivariate Polynomial Matrices 刘金旺 湖南科技大学 摘要: A multidimensional $(nD)$ system can be represented by a $nD$ polynomial matrix, we firstly investigate the general prime factorizations of $nD$ polynomial matrices: zero prime factorization, minor prime factorization, factor prime factorization. Second, we investigate the equivalence of $nD$ polynomial matrix. For a class of $nD$ polynomial matrices, giving some tractable necessary and sufficient conditions under which matrices are equivalent to their Smithform, there conditions are easily checked by computing the reduced Gr¨obner bases of some ideals. Finally, we investigate $nD$ polynomial matrix equivalence recursive problem, i.e., $\diag(1, B)\equiv\diag(1, 1, C)$, $B$ is or not equivalent to $diag(1, C1)$, a negative answer is obtained. 
On the connection between lexicographic Groebner bases and triangular sets 牟晨琪 北京航空航天大学 摘要: Lexicographic Groebner bases and triangular sets are standard tools in polynomial elimination theory. In this talk, we present new results on the intrinsic structures of lexicographic Greobner bases and relations between lexicographic Groenber bases and the minimal triangular sets contained in them called Wcharacteristic sets. It is shown that either this Wcharacteristic set is normal or there are explicit (pseudo)divisibility relations between the polynomials in it. Based on these properties of Wcharacteristic sets, we introduce the concept of characteristic pair consisting of a reduced lexicographic Groebner basis and its normal Wcharacteristic set, and design an algorithm for decomposing any polynomial set into finitely many characteristic pairs with associated zero relations, which provide representations of the zero of the polynomial set in terms of those of Groebner bases and in terms of those of triangular sets simultaneously. Several nice properties of the decomposition and the resulting characteristic pairs, in particular relationships between the Groebner basis and the Wcharacterisitic set in each pair, are established.

Real root classification and its applications 唐晓弦 Texas A&M University, USA 摘要: Classifying parameters according to numbers of real solutions of a general zerodimensional polynomial system is a fundamental problem in computational real algebraic geometry. In this talk, we give an introduction to a standard method of real root classification and many applications in system biology and statistics. 
Comprehensive Groebner System: Algorithms and Applications 王定康 中国科学院数学与系统科学研究院 摘要: We will introduce the definition of comprehensive Groebner System (CGS) and present some efficient algorithms to compute the CGS of a parametric polynomial system. We will also give some applications using CGS, including solving system of parametric polynomial equations, discovering geometric theorems automatically, performing quantifier elimination over an algebraic closed field and computing greatest common divisors of multivariate polynomials with parameters. 
Efficient Algorithms for Solving PolyPower Constraints 徐鸣 华东师范大学 摘要: We first consider a class of univariate real functionspolypowersthat extend integer exponents to
real algebraic exponents for polynomials. Our purpose is to isolate positive roots of such a function
into disjoint intervals, each contains exactly one positive root and together contain all, which
can be easily refined to any desired precision. To this end, we classify polypowers into simple
and nonsimple ones, depending on the number of linearly independent exponents. For the
former, based on GelfondSchneider theorem, we present two complete isolation algorithmsexclusion and differentiation. For the latter, their completeness depends on Schanuel's conjecture.

Safety Verification of Nonlinear Hybrid Systems Based on Bilinear Programming 杨争峰 华东师范大学 摘要: In safety verification of hybrid systems, barrier certificates are generated by solving the verification conditions derived from nonnegative representations of different types. In this talk, we present a new computational method, sequential linear programming projection, for directly solving the set of verification conditions represented by the KrivineVasilescuHandelman's positivstellensatz. The key idea is to decompose it into two successive optimization problems that refine the desired barrier certificate and those undetermined multipliers, respectively, and solve it in an iterative scheme. The most important benefit of the proposed approach lies in that it is much more effective than the LP relaxation method in producing real barrier certificates, and possesses a much lower computational complexity than the popular sum of square relaxation methods, which is demonstrated by the theoretical analysis on complexity and the experiment on a set of examples gathered from the literature. 