10:00am - 12:00pmAlgebraic methods for polynomial system solving solving
Chair(s): Mohab Safey El Din (Sorbonne Université, France), Éric Schost (University of Waterloo)
Polynomial system solving is at the heart of computational algebra and computational algebraic geometry. It arises in many applications ranging from computer security and coding theory (where computations must be done over finite fields) and engineering sciences such as chemistry, biology, signal theory or robotics among many others (here computations are done over inifinite domains such as complex or real numbers). The need of reliable algorithms for solving these problems is prominent because of the non-linear nature of the problems we have in hand.
Algebraic methods provide a nice framework for designing efficient and reliable algorithms solving polynoial systems. This mini-symposium will cover many aspects of this topic, including design of symbolic computation algorithms as well as the use of numerical methods in this framework with an emphasis on reliability.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
On polynomial and regular images of Euclidean spaces
José Fernando Galvan
Univ. Madrid
Let $f:=(f_1,ldots,f_m):R^ntoR^m$ be a map. We say that $f$ is em polynomial em if its components $f_k$ are polynomials. The map $f$ is em regular em if its components can be represented as quotients $f_k=frac{g_k}{h_k}$ of two polynomials $g_k,h_k$ such that $h_k$ never vanishes on $R^n$. More generally, the map $f$ is em Nash em if each component $f_k$ is a Nash function, that is, an analytic function whose graph is a {sl semialgebraic set}. Recall that a subset $SssubsetR^n$ is em semialgebraic em if it has a description as a finite boolean combination of polynomial equalities and inequalities. By Tarski-Seidenberg's principle the image of a map whose graph is a semialgebraic set is a semialgebraic set. Consequently, the images of polynomial, regular and Nash maps are semialgebraic sets. In 1990 {em Oberwolfach reelle algebraische Geometrie} week Gamboa proposed a kind of converse problem: em To characterize the semialgebraic sets in $R^m$ that are either polynomial or regular images of some $R^n$em. In the same period Shiota formulated a conjecture that characterizes Nash images of $R^n$, which has been recently proved by the author. In this talk we collect some of our main contributions to these problems and announce some future work concerning polynomial images of the unit closed ball. We have approached our contributions along the last two decades in three directions:
(i) To construct explicitly polynomial and regular maps whose images are the members of large families of semialgebraic sets whose boundaries are piecewise linear.
(ii) To find obstructions to be polynomial/regular images of $R^n$.
(iii) To prove Shiota's conjecture and some relevant consequences.
Degree bounds for the sparse Nullstellensatz
Gabriela Jeronimo
Univ. Buenos Aires
We will present new upper bounds both the degrees in Hilbert's Nullstellensatz and for the Noether exponent of a polynomial ideal in terms of mixed volumes of convex sets associated with the supports of a finite family of given generators. Our main results are the first upper bounds valid for arbitrary polynomial systems that distinguish the different supports of the polynomials. In the mixed sparse setting, they can be considerably smaller than previously known bounds. This is joint work with María Isabel Herrero and Juan Sabia (Universidad de Buenos Aires and CONICET, Argentina).
Signature-based Möller's algorithm for strong Gröbner bases over PIDs
Thibaut Verron
Johannes Kepler Univ.
Signature-based algorithms have become a standard approach for Gröbner basis computations for polynomial rings over fields, and recent work has focused on extending this technique to coefficients in rings.
Möller introduced in 1988 two algorithms for Gröbner bases over rings: one algorithm computing weak bases over any effective ring, and another computing strong bases if the coefficient ring is a Principal Ideal Domain (PID).
In 2018, we showed that it is possible to augment Möller's weak GB algorithm with signatures, in the case of PIDs. In this work, we show how the same technique can be used for Möller's strong GB algorithm. We prove that the resulting algorithm computes a strong Gröbner basis while ensuring that the signatures do not decrease, and in particular, that no signature drop occurs. Möller's strong GB algorithm requires special care compared to Möller's weak GB algorithm or to the fields case, because of its use of so-called $G$-polynomials whose signatures have to be controlled.
As in the case of fields or Möller's weak GB algorithm, it allows to use additional criteria such as the F5 criterion, which allows the algorithm to compute a Gröbner basis without a reduction to zero in the case of an ideal described by a regular sequence.
Furthermore, we show that Buchberger's coprime and chain criteria can be made compatible with signatures in Möller's strong GB algorithm. This makes use of the syzygy characterization of Gröbner bases, given by Möller's lifting theorem.
These results are supported by a toy implementation of the algorithms in Magma. In particular, Möller's strong GB algorithm does not suffer from the same combinatorial bottleneck as Möller's algorithm, which allowed us to gather experimental data regarding the number of $S$-polynomials computed, reduced and eliminated by each criterion.
(Joint work with Maria Francis)
Witness collections and a numerical algebraic geometry toolkit
Jose Rodriguez
Univ. of Wisconsin
A numerical description of an algebraic subvariety of projective space is given by a general linear section, called a witness set. For a subvariety of a product of projective spaces (a multiprojective variety), the corresponding numerical description is given by a witness set collection, whose structure is more involved. In this talk we build on recent work to give a complete treatment of witness set collections for multiprojective varieties, together with an algorithm for their numerical irreducible decomposition that exploits the structure of a witness set collection.