Computer Science

We provide two hybrid numeric-symbolic optimization algorithms, computing exact sums of nonnegative circuits (SONC) and sums of arithmetic-geometric-exponentials (SAGE) decompositions. Moreover, we provide a hybrid numeric-symbolic decision algorithm for polynomials lying in the interior of the SAGE cone. Each framework, inspired by previous contributions of Parrilo and Peyrl, is a rounding-projection procedure. For a polynomial lying in the interior of the SAGE cone, we prove that the decision algorithm terminates within a number of arithmetic operations, which is polynomial in the degree and number of terms of the input, and singly exponential in the number of variables. We also provide experimental comparisons regarding the implementation of the two optimization algorithms.

Given symmetric matrices A 0 , A 1 ,…, A n of size m with rational entries, the set of real vectors x=( x 1 ,…, x n ) such that the matrix A 0 + x 1 A 1 +⋯+ x n A n has non-negative eigenvalues is called a spectrahedron. Minimization of linear functions over spectrahedra is called semidefinite programming. Such problems appear frequently in control theory and real algebra, especially in the context of nonnegativity certificates for multivariate polynomials based on sums of squares. Numerical software for semidefinite programming are mostly based on interior point methods, assuming non-degeneracy properties such as the existence of an interior point in the spectrahedron. In this paper, we design an exact algorithm based on symbolic homotopy for solving semidefinite programs without assumptions on the feasible set, and we analyze its complexity. Because of the exactness of the output, it cannot compete with numerical routines in practice. However, we prove that solving such problems can be done in polynomial time if either n or m is fixed.

In this paper, we solve the existence problem of telescopers for rational functions in three discrete variables. We reduce the problem to that of deciding the summability of bivariate rational functions, which has been solved recently. The existence criteria we present is needed for detecting the termination of Zeilberger's algorithm to the function classes studied in this paper.

In the paper which inspired the SC-Square project, [E. Abraham, Building Bridges between Symbolic Computation and Satisfiability Checking, Proc. ISSAC '15, pp. 1-6, ACM, 2015] the author identified the use of sophisticated heuristics as a technique that the Satisfiability Checking community excels in and from which it is likely the Symbolic Computation community could learn and prosper. To start this learning process we summarise our experience with heuristic development for the computer algebra algorithm Cylindrical Algebraic Decomposition. We also propose and discuss standards and benchmarks as another area where Symbolic Computation could prosper from Satisfiability Checking expertise, noting that these have been identified as initial actions for the new SC-Square community in the CSA project, as described in [E.~Abraham et al., SC 2 : Satisfiability Checking meets Symbolic Computation (Project Paper)}, Intelligent Computer Mathematics (LNCS 9761), pp. 28--43, Springer, 2015].

Mathematical proofs are both paradigms of certainty and some of the most explicitly-justified arguments that we have in the cultural record. Their very explicitness, however, leads to a paradox, because their probability of error grows exponentially as the argument expands. Here we show that under a cognitively-plausible belief formation mechanism that combines deductive and abductive reasoning, mathematical arguments can undergo what we call an epistemic phase transition: a dramatic and rapidly-propagating jump from uncertainty to near-complete confidence at reasonable levels of claim-to-claim error rates. To show this, we analyze an unusual dataset of forty-eight machine-aided proofs from the formalized reasoning system Coq, including major theorems ranging from ancient to 21st Century mathematics, along with four hand-constructed cases from Euclid, Apollonius, Spinoza, and Andrew Wiles. Our results bear both on recent work in the history and philosophy of mathematics, and on a question, basic to cognitive science, of how we form beliefs, and justify them to others.

Two models were recently proposed to explore the robust hardness of Gröbner basis computation. Given a polynomial system, both models allow an algorithm to selectively ignore some of the polynomials: the algorithm is only responsible for returning a Gröbner basis for the ideal generated by the remaining polynomials. For the q -Fractional Gröbner Basis Problem the algorithm is allowed to ignore a constant (1−q) -fraction of the polynomials (subject to one natural structural constraint). Here we prove a new strongest-parameter result: even if the algorithm is allowed to choose a (3/10−ϵ) -fraction of the polynomials to ignore, and need only compute a Gröbner basis with respect to some lexicographic order for the remaining polynomials, this cannot be accomplished in polynomial time (unless P=NP ). This statement holds even if every polynomial has maximum degree 3. Next, we prove the first robust hardness result for polynomial systems of maximum degree 2: for the q -Fractional model a (1/5−ϵ) fraction of the polynomials may be ignored without losing provable NP-Hardness. Both theorems hold even if every polynomial contains at most three distinct variables. Finally, for the Strong c -partial Gröbner Basis Problem of De Loera et al. we give conditional results that depend on famous (unresolved) conjectures of Khot and Dinur, et al.

We extend the (continuous) multivariate Almkvist-Zeilberger algorithm in order to apply it for instance to special Feynman integrals emerging in renormalizable Quantum field Theories. We will consider multidimensional integrals over hyperexponential integrands and try to find closed form representations in terms of nested sums and products or iterated integrals. In addition, if we fail to compute a closed form solution in full generality, we may succeed in computing the first coefficients of the Laurent series expansions of such integrals in terms of indefinite nested sums and products or iterated integrals. In this article we present the corresponding methods and algorithms. Our Mathematica package MultiIntegrate, can be considered as an enhanced implementation of the (continuous) multivariate Almkvist Zeilberger algorithm to compute recurrences or differential equations for hyperexponential integrands and integrals. Together with the summation package Sigma and the package HarmonicSums our package provides methods to compute closed form representations (or coefficients of the Laurent series expansions) of multidimensional integrals over hyperexponential integrands in terms of nested sums or iterated integrals.

When studying local properties of a polynomial ideal, one usually needs a theoretic technique called localization. For most cases, in spite of its importance, the computation in a localized ring cannot be algorithmically preformed. On the other hand, the standard basis method is very effective for the computation in a special kind of localized rings, but for a general semigroup order the geometry of the localization of a positive-dimensional ideal is difficult to interpret. In this paper, we introduce a new ideal operation called extraction. For an ideal I in a polynomial ring K[ x 1 ,…, x n ] over a field K , we use another ideal J to control the primary components of I and the result β(I,J) is called the extraction of I by J . It is still a polynomial ideal and has a concrete geometric meaning in K ¯ n , i.e., we keep the branches of V(I)⊂ K ¯ n that intersect with V(J)⊂ K ¯ n and delete others, where K ¯ is the algebraic closure of K . This is what we mean by visible. On the other hand, we can use the standard basis method to compute a localized ideal corresponding to β(I,J) without a complete primary decomposition, and can do further computation in the localized ring such as determining the membership problem of β(I,J) . Moreover, we prove that extractions are as powerful as localizations in the sense that for any multiplicatively closed subset S of K[ x 1 ,…, x n ] and any polynomial ideal I , there always exists a polynomial ideal J such that β(I,J)=( S −1 I ) c .

In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial n'th Weyl algebra, the polynomial n'th shift algebra, and ZZ^n-graded polynomials in the n'th q-Weyl algebra. The most unexpected result is that this noncommutative problem of factoring partial differential operators can be approached effectively by reducing it to the problem of solving systems of polynomial equations over a commutative ring. In the case where a given polynomial is ZZ^n-graded, we can reduce the problem completely to factoring an element in a commutative multivariate polynomial ring. The implementation in Singular is effective on a broad range of polynomials and increases the ability of computer algebra systems to address this important problem. We compare the performance and output of our algorithm with other implementations in commodity computer algebra systems on nontrivial examples.

We discuss how to decide whether a given C-finite sequence can be written nontrivially as a product of two other C-finite sequences.