Computation of the Minimal Associated Primes

Solving systems of polynomial equations is a main task in Computer Algebra, although the precise meaning of what is an acceptable solution depends on the context. In this talk, we interpret it as finding the minimal associated primes of the ideal generated by the polynomials. Geometrically, this is equivalent to decompose the set of solutions into its irreducible components. We study the existing algorithms, and propose some modifications. A common technique used is to reduce the problem to the zero dimensional case. In a paper by Gianni, Trager and Zacharias they use this technique, combined with the splitting tool $I = (I : h^infty) cap langle I, h^m angle$ for some specific polynomial $h$ and integer $m$. This splitting introduces a number of redundant components that are not part of the original ideal. In the algorithm we present here, we use the reduction to the zero dimensional case, but we avoid working with the ideal $langle I, h^m angle$. As a result, when the ideal has components of different dimensions, our algorithm is usually more efficient

On the Pythagoras number for polynomials of degree 4 in 5 variables

We give an example of a polynomial of degree 4 in 5 variables that is the sum of squares of 8 polynomials and cannot be decomposed as the sum of 7 squares. This improves the current existing lower bound of 7 polynomials for the Pythagoras number $p(5,4)$

Strictly positive polynomials in the boundary of the SOS cone

We study the boundary of the cone of real polynomials that can be decomposed as a sum of squares (SOS) of real polynomials. This cone is included in the cone of nonnegative polynomials and both cones share a part of their boundary, which corresponds to polynomials that vanish at at least one point. We focus on the part of the boundary which is not shared, corresponding to strictly positive polynomials. For the cases of polynomials of degree 6 in 3 variables and degree 4 in 4 variables, this boundary has been completely characterized by G. Blekherman in a recent work. For the cases of more variables or higher degree, the problem is more complicated and very few examples or general results are known. Assuming a conjecture by D. Eisenbud, M. Green, and J. Harris, we obtain bounds for the maximum number of polynomials that can appear in a SOS decomposition and the maximum rank of the matrices in the Gram spectrahedron. In particular, for the case of homogeneous quartic polynomials in 5 variables, for which the required case of the conjecture has been recently proved, we obtain bounds that improve the general bounds known up to date. Finally, combining theoretical results with computational techniques, we find examples and counterexamples that allow us to better understand which of the results obtained by G. Blekherman can be extended to the general case and show that the bounds predicted by our results are attainable.Comment: 24 page

Parallel algorithms for normalization

Given a reduced affine algebra A over a perfect field K, we present parallel algorithms to compute the normalization \bar{A} of A. Our starting point is the algorithm of Greuel, Laplagne, and Seelisch, which is an improvement of de Jong's algorithm. First, we propose to stratify the singular locus Sing(A) in a way which is compatible with normalization, apply a local version of the normalization algorithm at each stratum, and find \bar{A} by putting the local results together. Second, in the case where K = Q is the field of rationals, we propose modular versions of the global and local-to-global algorithms. We have implemented our algorithms in the computer algebra system SINGULAR and compare their performance with that of the algorithm of Greuel, Laplagne, and Seelisch. In the case where K = Q, we also discuss the use of modular computations of Groebner bases, radicals, and primary decompositions. We point out that in most examples, the new algorithms outperform the algorithm of Greuel, Laplagne, and Seelisch by far, even if we do not run them in parallel.Comment: 19 page

Normalization of Rings

We present a new algorithm to compute the integral closure of a reduced Noetherian ring in its total ring of fractions. A modification, applicable in positive characteristic, where actually all computations are over the original ring, is also described. The new algorithm of this paper has been implemented in Singular, for localizations of affine rings with respect to arbitrary monomial orderings. Benchmark tests show that it is in general much faster than any other implementation of normalization algorithms known to us.Comment: Final version, to be published in JSC 201

Resolución de sistemas de ecuaciones polinomiales

Resolver sistemas de ecuaciones polinomiales en varias variables es un problema importante en álgebra computacional, con muchas aplicaciones. Cuando la cantidad de soluciones es infinita, no esta claro qué significa exactamente “resolver”. En esta charla, lo interpretamos como descomponer el conjunto de soluciones en sus componentes irreducibles. Algebraicamente, esto es equivalente a encontrar los primos minimales asociados al ideal generado por los polinomios. Veremos como se puede obtener algorítmicamente esta descomposicion y las herramientas que se necesitan, así como algunas aplicaciones concretas de estos algoritmos en robótica. Palabras clave: ecuaciones polinomiales, primos asociados, Groebner