Parallelization of Modular Algorithms

In this paper we investigate the parallelization of two modular algorithms. In fact, we consider the modular computation of Gr\"obner bases (resp. standard bases) and the modular computation of the associated primes of a zero-dimensional ideal and describe their parallel implementation in SINGULAR. Our modular algorithms to solve problems over Q mainly consist of three parts, solving the problem modulo p for several primes p, lifting the result to Q by applying Chinese remainder resp. rational reconstruction, and a part of verification. Arnold proved using the Hilbert function that the verification part in the modular algorithm to compute Gr\"obner bases can be simplified for homogeneous ideals (cf. \cite{A03}). The idea of the proof could easily be adapted to the local case, i.e. for local orderings and not necessarily homogeneous ideals, using the Hilbert-Samuel function (cf. \cite{Pf07}). In this paper we prove the corresponding theorem for non-homogeneous ideals in case of a global ordering.Comment: 16 page

Effective Differential Nullstellensatz for Ordinary DAE Systems with Constant Coefficients

We give upper bounds for the differential Nullstellensatz in the case of ordinary systems of differential algebraic equations over any field of constants $K$ of characteristic $0$. Let $\vec{x}$ be a set of $n$ differential variables, $\vec{f}$ a finite family of differential polynomials in the ring $K\{\vec{x}\}$ and $f\in K\{\vec{x}\}$ another polynomial which vanishes at every solution of the differential equation system $\vec{f}=0$ in any differentially closed field containing $K$. Let $d:=\max\{\deg(\vec{f}), \deg(f)\}$ and $\epsilon:=\max\{2,{\rm{ord}}(\vec{f}), {\rm{ord}}(f)\}$. We show that $f^M$ belongs to the algebraic ideal generated by the successive derivatives of $\vec{f}$ of order at most $L = (n\epsilon d)^{2^{c(n\epsilon)^3}}$, for a suitable universal constant $c>0$, and $M=d^{n(\epsilon +L+1)}$. The previously known bounds for $L$ and $M$ are not elementary recursive

Semidefinite Characterization and Computation of Real Radical Ideals

For an ideal $I\subseteq\mathbb{R}[x]$ given by a set of generators, a new semidefinite characterization of its real radical $I(V_\mathbb{R}(I))$ is presented, provided it is zero-dimensional (even if $I$ is not). Moreover we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety $V_\mathbb{R}(I)$ as well as a set of generators of the real radical ideal. The latter is obtained in the form of a border or Gr\"obner basis. The algorithm is based on moment relaxations and, in contrast to other existing methods, it exploits the real algebraic nature of the problem right from the beginning and avoids the computation of complex components.Comment: 41 page

On the Computation of Matrices of Traces and Radicals of Ideals

International audienceLet $f_1,\ldots,f_s \in \mathbb{K}[x_1,\ldots,x_m]$ be a system of polynomials generating a zero-dimensional ideal \I, where $\mathbb{K}$ is an arbitrary algebraically closed field. We study the computation of ``matrices of traces" for the factor algebra \A := \CC[x_1, \ldots , x_m]/ \I, i.e. matrices with entries which are trace functions of the roots of \I. Such matrices of traces in turn allow us to compute a system of multiplication matrices $\{M_{x_i}|i=1,\ldots,m\}$ of the radical \sqrt{\I}. We first propose a method using Macaulay type resultant matrices of $f_1,\ldots,f_s$ and a polynomial $J$ to compute moment matrices, and in particular matrices of traces for \A. Here $J$ is a polynomial generalizing the Jacobian. We prove bounds on the degrees needed for the Macaulay matrix in the case when \I has finitely many projective roots in \mathbb{P}^m_\CC. We also extend previous results which work only for the case where \A is Gorenstein to the non-Gorenstein case. The second proposed method uses Bezoutian matrices to compute matrices of traces of \A. Here we need the assumption that $s=m$ and $f_1,\ldots,f_m$ define an affine complete intersection. This second method also works if we have higher dimensional components at infinity. A new explicit description of the generators of \sqrt{\I} are given in terms of Bezoutians