Computer Science

Computations with integro-differential operators are often carried out in an associative algebra with unit, and they are essentially non-commutative computations. By adjoining a cocommutative co-product, one can have those operators perform act on a bialgebra isomorphic to an enveloping algebra. That gives an adequate framework for a computer-algebra implementation via monoidal factorization, (pure) transcendence bases and Poincaré--Birkhoff--Witt bases. In this paper, we systematically study these deformations, obtaining necessary and sufficient conditions for the operators to exist, and we give the most general cocommutative deformations of the shuffle co-product and an effective construction of pairs of bases in duality. The paper ends by the combinatorial setting of local systems of coordinates on the group of group-like series.

Signature-based algorithms are the latest and most efficient approach as of today to compute Gröbner bases for polynomial systems over fields. Recently, possible extensions of these techniques to general rings have attracted the attention of several authors. In this paper, we present a signature-based version of Möller's classical variant of Buchberger's algorithm for computing strong Gröbner bases over Principal Ideal Domains (or PIDs). It ensures that the signatures do not decrease during the algorithm, which makes it possible to apply classical signature criteria for further optimization. In particular, with the F5 criterion, the signature version of Möller's algorithm computes a Gröbner basis without reductions to zero for a polynomial system given by a regular sequence. We also show how Buchberger's chain criterion can be implemented so as to be compatible with the signatures. We prove correctness and termination of the algorithm. Furthermore, we have written a toy implementation in Magma, allowing us to quantitatively compare the efficiency of the various criteria for eliminating S-pairs.

We consider the problem of determining multiple steady states for positive real values in models of biological networks. Investigating the potential for these in models of the mitogen-activated protein kinases (MAPK) network has consumed considerable effort using special insights into the structure of corresponding models. Here we apply combinations of symbolic computation methods for mixed equality/inequality systems, specifically virtual substitution, lazy real triangularization and cylindrical algebraic decomposition. We determine multistationarity of an 11-dimensional MAPK network when numeric values are known for all but potentially one parameter. More precisely, our considered model has 11 equations in 11 variables and 19 parameters, 3 of which are of interest for symbolic treatment, and furthermore positivity conditions on all variables and parameters.

We consider the problem of finding a condition for a univariate polynomial having a given multiplicity structure when the number of distinct roots is given. It is well known that such conditions can be written as conjunctions of several polynomial equations and one inequation in the coefficients, by using repeated parametric gcd's. In this paper, we give a novel condition which is not based on repeated gcd's. Furthermore, it is shown that the number of polynomials in the condition is optimal and the degree of polynomials is smaller than that in the previous condition based on repeated gcd's.

A summation framework is developed that enhances Karr's difference field approach. It covers not only indefinite nested sums and products in terms of transcendental extensions, but it can treat, e.g., nested products defined over roots of unity. The theory of the so-called RΠ Σ ∗ -extensions is supplemented by algorithms that support the construction of such difference rings automatically and that assist in the task to tackle symbolic summation problems. Algorithms are presented that solve parameterized telescoping equations, and more generally parameterized first-order difference equations, in the given difference ring. As a consequence, one obtains algorithms for the summation paradigms of telescoping and Zeilberger's creative telescoping. With this difference ring theory one obtains a rigorous summation machinery that has been applied to numerous challenging problems coming, e.g., from combinatorics and particle physics.

For linear recurrence systems, the problem of finding rational solutions is reduced to the problem of computing polynomial solutions by computing a content bound or a denominator bound. There are several bounds in the literature. The sharpest bound leads to polynomial solutions of lower degrees, but this advantage need not compensate for the time spent on computing that bound. To strike the best balance between sharpness of the bound versus CPU time spent obtaining it, we will give a family of bounds. The J 'th member of this family is similar to (Abramov, Barkatou, 1998) when J=1 , similar to (van Hoeij, 1998) when J is large, and novel for intermediate values of J , which give the best balance between sharpness and CPU time. The setting for our content bounds are systems τ(Y)=MY where τ is an automorphism of a UFD, and M is an invertible matrix with entries in its field of fractions. This setting includes the shift case, the q -shift case, the multi-basic case and others. We give two versions, a global version, and a version that bounds each entry separately.

Let P and Q be two polynomials in K[x, y] with degree at most d, where K is a field. Denoting by R ∈ K[x] the resultant of P and Q with respect to y, we present an algorithm to compute R mod x^k in O~(kd) arithmetic operations in K, where the O~ notation indicates that we omit polylogarithmic factors. This is an improvement over state-of-the-art algorithms that require to compute R in O~(d^3) operations before computing its first k coefficients.

We design an algorithm for computing the p -curvature of a differential system in positive characteristic p . For a system of dimension r with coefficients of degree at most d , its complexity is $\softO (p d r^\omega)$ operations in the ground field (where ω denotes the exponent of matrix multiplication), whereas the size of the output is about pd r 2 . Our algorithm is then quasi-optimal assuming that matrix multiplication is (\emph{i.e.} ω=2 ). The main theoretical input we are using is the existence of a well-suited ring of series with divided powers for which an analogue of the Cauchy--Lipschitz Theorem holds.

We propose a probabilistic variant of Brill-Noether's algorithm for computing a basis of the Riemann-Roch space L(D) associated to a divisor D on a projective nodal plane curve C over a sufficiently large perfect field k . Our main result shows that this algorithm requires at most O(max(deg(C ) 2ω ,deg( D + ) ω )) arithmetic operations in k , where ω is a feasible exponent for matrix multiplication and D + is the smallest effective divisor such that D + ≥D . This improves the best known upper bounds on the complexity of computing Riemann-Roch spaces. Our algorithm may fail, but we show that provided that a few mild assumptions are satisfied, the failure probability is bounded by O(max(deg(C ) 4 ,deg( D + ) 2 )/|E|) , where E is a finite subset of k in which we pick elements uniformly at random. We provide a freely available C++/NTL implementation of the proposed algorithm and we present experimental data. In particular, our implementation enjoys a speedup larger than 6 on many examples (and larger than 200 on some instances over large finite fields) compared to the reference implementation in the Magma computer algebra system. As a by-product, our algorithm also yields a method for computing the group law on the Jacobian of a smooth plane curve of genus g within O( g ω ) operations in k , which equals the best known complexity for this problem.

We generalize the framework of virtual substitution for real quantifier elimination to arbitrary but bounded degrees. We make explicit the representation of test points in elimination sets using roots of parametric univariate polynomials described by Thom codes. Our approach follows an early suggestion by Weispfenning, which has never been carried out explicitly. Inspired by virtual substitution for linear formulas, we show how to systematically construct elimination sets containing only test points representing lower bounds.