8,866 research outputs found
Solving multivariate polynomial systems and an invariant from commutative algebra
The complexity of computing the solutions of a system of multivariate
polynomial equations by means of Gr\"obner bases computations is upper bounded
by a function of the solving degree. In this paper, we discuss how to
rigorously estimate the solving degree of a system, focusing on systems arising
within public-key cryptography. In particular, we show that it is upper bounded
by, and often equal to, the Castelnuovo Mumford regularity of the ideal
generated by the homogenization of the equations of the system, or by the
equations themselves in case they are homogeneous. We discuss the underlying
commutative algebra and clarify under which assumptions the commonly used
results hold. In particular, we discuss the assumption of being in generic
coordinates (often required for bounds obtained following this type of
approach) and prove that systems that contain the field equations or their fake
Weil descent are in generic coordinates. We also compare the notion of solving
degree with that of degree of regularity, which is commonly used in the
literature. We complement the paper with some examples of bounds obtained
following the strategy that we describe
Using Elimination Theory to construct Rigid Matrices
The rigidity of a matrix A for target rank r is the minimum number of entries
of A that must be changed to ensure that the rank of the altered matrix is at
most r. Since its introduction by Valiant (1977), rigidity and similar
rank-robustness functions of matrices have found numerous applications in
circuit complexity, communication complexity, and learning complexity. Almost
all nxn matrices over an infinite field have a rigidity of (n-r)^2. It is a
long-standing open question to construct infinite families of explicit matrices
even with superlinear rigidity when r = Omega(n).
In this paper, we construct an infinite family of complex matrices with the
largest possible, i.e., (n-r)^2, rigidity. The entries of an n x n matrix in
this family are distinct primitive roots of unity of orders roughly exp(n^2 log
n). To the best of our knowledge, this is the first family of concrete (but not
entirely explicit) matrices having maximal rigidity and a succinct algebraic
description.
Our construction is based on elimination theory of polynomial ideals. In
particular, we use results on the existence of polynomials in elimination
ideals with effective degree upper bounds (effective Nullstellensatz). Using
elementary algebraic geometry, we prove that the dimension of the affine
variety of matrices of rigidity at most k is exactly n^2-(n-r)^2+k. Finally, we
use elimination theory to examine whether the rigidity function is
semi-continuous.Comment: 25 Pages, minor typos correcte
Exploiting chordal structure in polynomial ideals: a Gr\"obner bases approach
Chordal structure and bounded treewidth allow for efficient computation in
numerical linear algebra, graphical models, constraint satisfaction and many
other areas. In this paper, we begin the study of how to exploit chordal
structure in computational algebraic geometry, and in particular, for solving
polynomial systems. The structure of a system of polynomial equations can be
described in terms of a graph. By carefully exploiting the properties of this
graph (in particular, its chordal completions), more efficient algorithms can
be developed. To this end, we develop a new technique, which we refer to as
chordal elimination, that relies on elimination theory and Gr\"obner bases. By
maintaining graph structure throughout the process, chordal elimination can
outperform standard Gr\"obner basis algorithms in many cases. The reason is
that all computations are done on "smaller" rings, of size equal to the
treewidth of the graph. In particular, for a restricted class of ideals, the
computational complexity is linear in the number of variables. Chordal
structure arises in many relevant applications. We demonstrate the suitability
of our methods in examples from graph colorings, cryptography, sensor
localization and differential equations.Comment: 40 pages, 5 figure
Gr\"obner Bases of Bihomogeneous Ideals generated by Polynomials of Bidegree (1,1): Algorithms and Complexity
Solving multihomogeneous systems, as a wide range of structured algebraic
systems occurring frequently in practical problems, is of first importance.
Experimentally, solving these systems with Gr\"obner bases algorithms seems to
be easier than solving homogeneous systems of the same degree. Nevertheless,
the reasons of this behaviour are not clear. In this paper, we focus on
bilinear systems (i.e. bihomogeneous systems where all equations have bidegree
(1,1)). Our goal is to provide a theoretical explanation of the aforementionned
experimental behaviour and to propose new techniques to speed up the Gr\"obner
basis computations by using the multihomogeneous structure of those systems.
The contributions are theoretical and practical. First, we adapt the classical
F5 criterion to avoid reductions to zero which occur when the input is a set of
bilinear polynomials. We also prove an explicit form of the Hilbert series of
bihomogeneous ideals generated by generic bilinear polynomials and give a new
upper bound on the degree of regularity of generic affine bilinear systems.
This leads to new complexity bounds for solving bilinear systems. We propose
also a variant of the F5 Algorithm dedicated to multihomogeneous systems which
exploits a structural property of the Macaulay matrix which occurs on such
inputs. Experimental results show that this variant requires less time and
memory than the classical homogeneous F5 Algorithm.Comment: 31 page
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