Computer Science

Inspired by Faugère and Mou's sparse FGLM algorithm, we show how using linear recurrent multi-dimensional sequences can allow one to perform operations such as the primary decomposition of an ideal, by computing the annihilator of one or several such sequences.

Using integration by parts relations, Feynman integrals can be represented in terms of coupled systems of differential equations. In the following we suppose that the unknown Feynman integrals can be given in power series representations, and that sufficiently many initial values of the integrals are given. Then there exist algorithms that decide constructively if the coefficients of their power series representations can be given within the class of nested sums over hypergeometric products. In this article we will work out the calculation steps that solve this problem. First, we will present a successful tactic that has been applied recently to challenging problems coming from massive 3-loop Feynman integrals. Here our main tool is to solve scalar linear recurrences within the class of nested sums over hypergeometric products. Second, we will present a new variation of this tactic which relies on more involved summation technologies but succeeds in reducing the problem to solve scalar recurrences with lower recurrence orders. The article will work out the different challenges of this new tactic and demonstrates how they can be treated efficiently with our existing summation technologies.

We describe the Aligator.jl software package for automatically generating all polynomial invariants of the rich class of extended P-solvable loops with nested conditionals. Aligator.jl is written in the programming language Julia and is open-source. Aligator.jl transforms program loops into a system of algebraic recurrences and implements techniques from symbolic computation to solve recurrences, derive closed form solutions of loop variables and infer the ideal of polynomial invariants by variable elimination based on Gröbner basis computation.

We consider the additive decomposition problem in primitive towers and present an algorithm to decompose a function in an S-primitive tower as a sum of a derivative in the tower and a remainder which is minimal in some sense. Special instances of S-primitive towers include differential fields generated by finitely many logarithmic functions and logarithmic integrals. A function in an S-primitive tower is integrable in the tower if and only if the remainder is equal to zero. The additive decomposition is achieved by viewing our towers not as a traditional chain of extension fields, but rather as a direct sum of certain subrings. Furthermore, we can determine whether or not a function in an S-primitive tower has an elementary integral without solving any differential equations. We also show that a kind of S-primitive towers, known as logarithmic towers, can be embedded into a particular extension where we can obtain a finer remainder.

Quorum systems are a key mathematical abstraction in distributed fault-tolerant computing for capturing trust assumptions. A quorum system is a collection of subsets of all processes, called quorums, with the property that each pair of quorums have a non-empty intersection. They can be found at the core of many reliable distributed systems, such as cloud computing platforms, distributed storage systems and blockchains. In this paper we give a new interpretation of quorum systems, starting with classical majority-based quorum systems and extending this to Byzantine quorum systems. We propose an algebraic representation of the theory underlying quorum systems making use of multivariate polynomial ideals, incorporating properties of these systems, and studying their algebraic varieties. To achieve this goal we will exploit properties of Boolean Groebner bases. The nice nature of Boolean Groebner bases allows us to avoid part of the combinatorial computations required to check consistency and availability of quorum systems. Our results provide a novel approach to test quorum systems properties from both algebraic and algorithmic perspectives.

An algorithm to generate a minimal comprehensive Gröbner\, basis of a parametric polynomial system from an arbitrary faithful comprehensive Gröbner\, system is presented. A basis of a parametric polynomial ideal is a comprehensive Gröbner\, basis if and only if for every specialization of parameters in a given field, the specialization of the basis is a Gröbner\, basis of the associated specialized polynomial ideal. The key idea used in ensuring minimality is that of a polynomial being essential with respect to a comprehensive Gröbner\, basis. The essentiality check is performed by determining whether a polynomial can be covered for various specializations by other polynomials in the associated branches in a comprehensive Gröbner\, system. The algorithm has been implemented and successfully tried on many examples from the literature.

Let Δ x f(x,y)=f(x+1,y)−f(x,y) and Δ y f(x,y)=f(x,y+1)−f(x,y) be the difference operators with respect to x and y . A rational function f(x,y) is called summable if there exist rational functions g(x,y) and h(x,y) such that f(x,y)= Δ x g(x,y)+ Δ y h(x,y) . Recently, Chen and Singer presented a method for deciding whether a rational function is summable. To implement their method in the sense of algorithms, we need to solve two problems. The first is to determine the shift equivalence of two bivariate polynomials. We solve this problem by presenting an algorithm for computing the dispersion sets of any two bivariate polynomials. The second is to solve a univariate difference equation in an algebraically closed field. By considering the irreducible factorization of the denominator of f(x,y) in a general field, we present a new criterion which requires only finding a rational solution of a bivariate difference equation. This goal can be achieved by deriving a universal denominator of the rational solutions and a degree bound on the numerator. Combining these two algorithms, we can decide the summability of a bivariate rational function.

Computing a basis for the exponent lattice of algebraic numbers is a basic problem in the field of computational number theory with applications to many other areas. The main cost of a well-known algorithm \cite{ge1993algorithms,kauers2005algorithms} solving the problem is on computing the primitive element of the extended field generated by the given algebraic numbers. When the extended field is of large degree, the problem seems intractable by the tool implementing the algorithm. In this paper, a special kind of exponent lattice basis is introduced. An important feature of the basis is that it can be inductively constructed, which allows us to deal with the given algebraic numbers one by one when computing the basis. Based on this, an effective framework for constructing exponent lattice basis is proposed. Through computing a so-called pre-basis first and then solving some linear Diophantine equations, the basis can be efficiently constructed. A new certificate for multiplicative independence and some techniques for decreasing degrees of algebraic numbers are provided to speed up the computation. The new algorithm has been implemented with Mathematica and its effectiveness is verified by testing various examples. Moreover, the algorithm is applied to program verification for finding invariants of linear loops.

There are several efficient methods to solve linear interval polynomial systems in the context of interval computations, however, the general case of interval polynomial systems is not yet covered as well. In this paper we introduce a new elimination method to solve and analyse interval polynomial systems, in general case. This method is based on computational algebraic geometry concepts such as polynomial ideals and Groebner basis computation. Specially, we use the comprehensive Groebner system concept to keep the dependencies between interval coefficients. At the end of paper, we will state some applications of our method to evaluate its performance.

The Truncated Fourier Transform (TFT) is a variation of the Discrete Fourier Transform (DFT/FFT) that allows for input vectors that do NOT have length 2 n for n a positive integer. We present the univariate version of the TFT, originally due to Joris van der Hoeven, heavily illustrating the presentation in order to make these methods accessible to a broader audience.