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

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

Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization

The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear algebra or semidefinite programming relaxations of many kinds of feasibility or optimization questions. We are particularly interested in problems arising in combinatorial optimization.Comment: 28 pages, survey pape

Solving Degenerate Sparse Polynomial Systems Faster

Consider a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic. We present a fast method to find a point in every irreducible component of the zero set Z of F. Our techniques allow us to sharpen and lower prior complexity bounds for this problem by fully taking into account the monomial term structure. As a corollary of our development we also obtain new explicit formulae for the exact number of isolated roots of F and the intersection multiplicity of the positive-dimensional part of Z. Finally, we present a combinatorial construction of non-degenerate polynomial systems, with specified monomial term structure and maximally many isolated roots, which may be of independent interest.Comment: This is the final journal version of math.AG/9702222 (``Toric Generalized Characteristic Polynomials''). This final version is a major revision with several new theorems, examples, and references. The prior results are also significantly improve

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

Stable normal forms for polynomial system solving

This paper describes and analyzes a method for computing border bases of a zero-dimensional ideal $I$. The criterion used in the computation involves specific commutation polynomials and leads to an algorithm and an implementation extending the one provided in [MT'05]. This general border basis algorithm weakens the monomial ordering requirement for \grob bases computations. It is up to date the most general setting for representing quotient algebras, embedding into a single formalism Gr\"obner bases, Macaulay bases and new representation that do not fit into the previous categories. With this formalism we show how the syzygies of the border basis are generated by commutation relations. We also show that our construction of normal form is stable under small perturbations of the ideal, if the number of solutions remains constant. This new feature for a symbolic algorithm has a huge impact on the practical efficiency as it is illustrated by the experiments on classical benchmark polynomial systems, at the end of the paper