Computer Science

Multi-homogeneous polynomial systems arise in many applications. We provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input system under some genericity assumptions. The assumptions essentially imply that the Jacobian matrix of the system under study has maximal rank at the solution set and that this solution set if finite. The algorithm is probabilistic and a probability analysis is provided. Next, we apply these results to the problem of optimizing a linear map on the real trace of an algebraic set. Under some genericity assumptions, we provide bit complexity estimates for solving this polynomial minimization problem.

Abramov's algorithm enables us to decide whether a univariate rational function can be written as a difference of another rational function, which has been a fundamental algorithm for rational summation. In 2014, Chen and Singer generalized Abramov's algorithm to the case of rational functions in two ( q -)discrete variables. In this paper we solve the remaining three mixed cases, which completes our recent project on bivariate extensions of Abramov's algorithm for rational summation.

A widely used method for solving SOS (Sum Of Squares) decomposition problem is to reduce it to the problem of semi-definite programs (SDPs) which can be efficiently solved in theory. In practice, although many SDP solvers can work out some problems of big scale, the efficiency and reliability of such method decrease greatly while the input size increases. Recently, by exploiting the sparsity of the input SOS decomposition problem, some preprocessing algorithms were proposed [5,17], which first divide the input problem satisfying special definitions or properties into smaller SDP problems and then pass the smaller ones to SDP solvers to obtain reliable results efficiently. A natural question is that to what extent the above mentioned preprocessing algorithms work. That is, how many polynomials satisfying those definitions or properties are there in the SOS polynomials? In this paper, we define a concept of block SOS decomposable polynomials which is a generalization of those special classes in [5] and [17]. Roughly speaking, it is a class of polynomials whose SOS decomposition problem can be transformed into smaller ones (in other words, the corresponding SDP matrices can be block-diagnolized) by considering their supports only (coefficients are not considered). Then we prove that the set of block SOS decomposable polynomials has measure zero in the set of SOS polynomials. That means if we only consider supports (not with coefficients) of polynomials, such algorithms decreasing the size of SDPs for those SDP-based SOS solvers can only work on very few polynomials. As a result, this shows that the SOS decomposition problems that can be optimized by the above mentioned preprocessing algorithms are very few.

Consider a zero-dimensional ideal I in K[ X 1 ,…, X n ] . Inspired by Faugère and Mou's Sparse FGLM algorithm, we use Krylov sequences based on multiplication matrices of I in order to compute a description of its zero set by means of univariate polynomials. Steel recently showed how to use Coppersmith's block-Wiedemann algorithm in this context; he describes an algorithm that can be easily parallelized, but only computes parts of the output in this manner. Using generating series expressions going back to work of Bostan, Salvy, and Schost, we show how to compute the entire output for a small overhead, without making any assumption on the ideal I other than it having dimension zero. We then propose a refinement of this idea that partially avoids the introduction of a generic linear form. We comment on experimental results obtained by an implementation based on the C++ libraries Eigen, LinBox and NTL.

We look at Bohemian matrices, specifically those with entries from {−1,0,+1} . More, we specialize the matrices to be upper Hessenberg, with subdiagonal entries ±1 . Many properties remain after these specializations, some of which surprised us. We find two recursive formulae for the characteristic polynomials of upper Hessenberg matrices. Focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give a lower bound on their height. This bound is exponential in the order of the matrix. We count stable matrices, normal matrices, and neutral matrices, and tabulate the results of our experiments. We prove a theorem about the only possible kinds of normal matrices amongst a specific family of Bohemian upper Hessenberg matrices.

We look at Bohemian matrices, specifically those with entries from {−1,0,+1} . More, we specialize the matrices to be upper Hessenberg, with subdiagonal entries 1 . Even more, we consider Toeplitz matrices of this kind. Many properties remain after these specializations, some of which surprised us. Focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give a lower bound on their height. This bound is exponential in the order of the matrix.

The theory of border bases for zero-dimensional ideals has attracted several researchers in symbolic computation due to their numerical stability and mathematical elegance. As shown in (Francis & Dukkipati, J. Symb. Comp., 2014), one can extend the concept of border bases over Noetherian rings whenever the corresponding residue class ring is finitely generated and free. In this paper we address the following problem: Can the concept of border basis over Noetherian rings exists for ideals when the corresponding residue class rings are finitely generated but need not necessarily be free modules? We present a border division algorithm and prove the termination of the algorithm for a special class of border bases. We show the existence of such border bases over Noetherian rings and present some characterizations in this regard. We also show that certain reduced Gröbner bases over Noetherian rings are contained in this class of border bases.

Normalization of polynomials plays an essential role in the approximate basis computation of vanishing ideals. In computer algebra, coefficient normalization, which normalizes a polynomial by its coefficient norm, is the most common method. In this study, we propose gradient-weighted normalization for the approximate border basis computation of vanishing ideals, inspired by the recent results in machine learning. The data-dependent nature of gradient-weighted normalization leads to powerful properties such as better stability against perturbation and consistency in the scaling of input points, which cannot be attained by the conventional coefficient normalization. With a slight modification, the analysis of algorithms with coefficient normalization still works with gradient-weighted normalization and the time complexity does not change. We also provide an upper bound on the coefficient norm based on the gradient-weighted norm, which allows us to discuss the approximate border bases with gradient-weighted normalization from the perspective of the coefficient norm.

We provide bounds on the size of operators obtained by algorithms for executing D-finite closure properties. For operators of small order, we give bounds on the degree and on the height (bit-size). For higher order operators, we give degree bounds that are parameterized with respect to the order and reflect the phenomenon that higher order operators may have lower degrees (order-degree curves).

It is well known that the composition of a D-finite function with an algebraic function is again D-finite. We give the first estimates for the orders and the degrees of annihilating operators for the compositions. We find that the analysis of removable singularities leads to an order-degree curve which is much more accurate than the order-degree curve obtained from the usual linear algebra reasoning.