Systems of polynomial equations often have symmetries. In solving such a system using Buchberger's algorithm, the symmetries are neglected. Incorporating symmetries into the solution process enables us to solve larger problems than with Buchberger's algorithm alone. This paper presents a method that shows how this can be achieved and also gives an algorithm that brings together continuously par...

This paper presents a new algorithm that computes the local algebras of the roots of a zero-dimensional polynomial equation system, with a number of operations in the coefficient field that is polynomial in the number of variables, in the evaluation cost of the equations and in a Bézout number.

The use of algebraic techniques to solve combinatorial problems is studied in this paper. We formulate the rainbow connectivity problem as a system of polynomial equations. We first consider the case of two colors for which the problem is known to be hard and we then extend the approach to the general case. We also give a formulation of the rainbow connectivity problem as an ideal membership pr...

Sudoku is a logic-based placement puzzle. We recall how to translate this puzzle into a 9-colouring problem which is equivalent to a (big) algebraic system of polynomial equations. We study how far Gröbner bases techniques can be used to treat these systems produced by Sudokus. This general purpose tool can not be considered as a good solver, but we show that it can be useful to provide informa...

We study geometric properties of the solution variety for the problem of approximating solutions of systems of polynomial equations. We prove that given two pairs (fi, ζi), i = 1, 2, there exists a short path joining them such that the complexity of following the path is bounded by the logarithm of the condition number of the problems.

We study the algorithmic complexity of determining whether a system of polynomial equations over a finite algebra admits a solution. We prove that the problem has a dichotomy in the class of finite groupoids with an identity element. By developing the underlying idea further, we present a dichotomy theorem in the class of finite algebras that admit a non-trivial idempotent Maltsev condition. Th...

Given convex polytopes P1, . . . , Pr ⊂ R n and finite subsets WI of the Minkowsky sums PI = � i∈I Pi, we consider the quantityN(W) = � I⊂[r] (−1) r−|I|� WI � �. If WI = Z n ∩ PI and P1, . . . , Pn are lattice polytopes in R , then N(W) is the classical mixed volume of P1, . . . , Pn giving the number of complex solutions of a general complex polynomial system with Newton polytopes P1, . . . , ...

In this paper we describe how to improve the performance of the symbolicnumeric method in [19, 20] for computing the multiplicity structure and refining approximate isolated singular solutions in the breadth-one case. By introducing a parameterized deflated system with smoothing parameters, we generalize the algorithm in [33] to compute verified error bounds such that a slightly perturbed polyn...

We use Bernstein’s Theorem [1] to obtain combinatorial bounds for the number of embeddings of Laman graph frameworks modulo rigid motions. For this, we study the mixed volume of suitable systems of polynomial equations obtained from the edge length constraints. The bounds can easily be computed and for some classes of graphs, the bounds are tight.

We study Cesàro (C, δ) means for two-variable Jacobi polynomials on the parabolic biangle B = {(x1, x2) ∈ R2 : 0 ≤ x1 ≤ x2 ≤ 1}. Using the product formula derived by Koornwinder & Schwartz for this polynomial system, the Cesàro operator can be interpreted as a convolution operator. We then show that the Cesàro (C, δ) means of the orthogonal expansion on the biangle are uniformly bounded if δ > ...