Computer Science

Multiple binomial sums form a large class of multi-indexed sequences, closed under partial summation, which contains most of the sequences obtained by multiple summation of products of binomial coefficients and also all the sequences with algebraic generating function. We study the representation of the generating functions of binomial sums by integrals of rational functions. The outcome is twofold. Firstly, we show that a univariate sequence is a multiple binomial sum if and only if its generating function is the diagonal of a rational function. Secondly, we propose algorithms that decide the equality of multiple binomial sums and that compute recurrence relations for them. In conjunction with geometric simplifications of the integral representations, this approach behaves well in practice. The process avoids the computation of certificates and the problem of the appearance of spurious singularities that afflicts discrete creative telescoping, both in theory and in practice.

We show a new algorithm and its implementation for multiplying bit-polynomials of large degrees. The algorithm is based on evaluating polynomials at a specific set comprising a natural set for evaluation with additive FFT and a high order element under Frobenius map of F 2 . With the high order element, we can derive more values of the polynomials under Frobenius map. Besides, we also adapt the additive FFT to efficiently evaluate polynomials at the set with an encoding process. For the implementation, we reorder the computations in the additive FFT for reducing the number of memory writes and hiding the latency for reads. The algebraic operations, including field multiplication, bit-matrix transpose, and bit-matrix multiplication, are implemented with efficient SIMD instructions. As a result, we effect a software of best known efficiency, shown in our experiments.

We present a package to perform partial fraction decompositions of multivariate rational functions. The algorithm allows to systematically avoid spurious denominator factors and is capable of producing unique results also when being applied to terms of a sum separately. The package is designed to work in Mathematica, but also provides interfaces to the Form and Singular computer algebra systems.

We present a method for solving polynomial equations over idempotent omega-continuous semirings. The idea is to iterate over the semiring of functions rather than the semiring of interest, and only evaluate when needed. The key operation is substitution. In the initial step, we compute a linear completion of the system of equations that exhaustively inserts the equations into one another. With functions as approximants, the following steps insert the current approximant into itself. Since the iteration improves its precision by substitution rather than computation we named it Munchausen, after the fictional baron that pulled himself out of a swamp by his own hair. The first result shows that an evaluation of the n-th Munchausen approximant coincides with the 2^n-th Newton approximant. Second, we show how to compute linear completions with standard techniques from automata theory. In particular, we are not bound to (but can use) the notion of differentials prominent in Newton iteration.

We estimate the Boolean complexity of multiplication of structured matrices by a vector and the solution of nonsingular linear systems of equations with these matrices. We study four basic most popular classes, that is, Toeplitz, Hankel, Cauchy and Van-der-monde matrices, for which the cited computational problems are equivalent to the task of polynomial multiplication and division and polynomial and rational multipoint evaluation and interpolation. The Boolean cost estimates for the latter problems have been obtained by Kirrinnis in \cite{kirrinnis-joc-1998}, except for rational interpolation, which we supply now. All known Boolean cost estimates for these problems rely on using Kronecker product. This implies the d -fold precision increase for the d -th degree output, but we avoid such an increase by relying on distinct techniques based on employing FFT. Furthermore we simplify the analysis and make it more transparent by combining the representation of our tasks and algorithms in terms of both structured matrices and polynomials and rational functions. This also enables further extensions of our estimates to cover Trummer's important problem and computations with the popular classes of structured matrices that generalize the four cited basic matrix classes.

In the sparse polynomial multiplication problem, one is asked to multiply two sparse polynomials f and g in time that is proportional to the size of the input plus the size of the output. The polynomials are given via lists of their coefficients F and G, respectively. Cole and Hariharan (STOC 02) have given a nearly optimal algorithm when the coefficients are positive, and Arnold and Roche (ISSAC 15) devised an algorithm running in time proportional to the "structural sparsity" of the product, i.e. the set supp(F)+supp(G). The latter algorithm is particularly efficient when there not "too many cancellations" of coefficients in the product. In this work we give a clean, nearly optimal algorithm for the sparse polynomial multiplication problem.

Polynomial Systems, or at least their algorithms, have the reputation of being doubly-exponential in the number of variables [Mayr and Mayer, 1982], [Davenport and Heintz, 1988]. Nevertheless, the Bezout bound tells us that that number of zeros of a zero-dimensional system is singly-exponential in the number of variables. How should this contradiction be reconciled? We first note that [Mayr and Ritscher, 2013] shows that the doubly exponential nature of Gröbner bases is with respect to the dimension of the ideal, not the number of variables. This inspires us to consider what can be done for Cylindrical Algebraic Decomposition which produces a doubly-exponential number of polynomials of doubly-exponential degree. We review work from ISSAC 2015 which showed the number of polynomials could be restricted to doubly-exponential in the (complex) dimension using McCallum's theory of reduced projection in the presence of equational constraints. We then discuss preliminary results showing the same for the degree of those polynomials. The results are under primitivity assumptions whose importance we illustrate.

Based on a modified version of Abramov-Petkovšek reduction, a new algorithm to compute minimal telescopers for bivariate hypergeometric terms was developed last year. We investigate further in this paper and present a new argument for the termination of this algorithm, which provides an independent proof of the existence of telescopers and even enables us to derive lower as well as upper bounds for the order of telescopers for hypergeometric terms. Compared to the known bounds in the literature, our bounds are sometimes better, and never worse than the known ones.

In certain scientific domains, there is a need for tensor operations. To facilitate tensor computations,computer algebra systems are employed. In our research, we have been using Cadabra as the main computer algebra system for several years. Recently, an operable second version of this software was released. In this version, a number of improvements were made that can be regarded as revolutionary ones. The most significant improvements are the implementation of component computations and the change in the ideology of the Cadabra's software mechanism as compared to the first version. This paper provides a brief overview of the key improvements in the Cadabra system.

The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?" The question has been asked before but as yet the leading algorithms for answering such questions have not been formalised. We present a thesis that a new algorithm for ascertaining satisfiability of formulae over the reals via Cylindrical Algebraic Coverings [Ábrahám, Davenport, England, Kremer, \emph{Deciding the Consistency of Non-Linear Real Arithmetic Constraints with a Conflict Driver Search Using Cylindrical Algebraic Coverings}, 2020] might provide trace and outputs that allow the results to be more susceptible to machine verification than those of competing algorithms.