Ioannis Z. Emiris

Efficient incremental algorithms for the sparse resultant and the mixed volume

Abstract We propose a new and efficient algorithm for computing the sparse resultant of a system of n + 1 polynomial equations in n unknowns. This algorithm produces a matrix whose entries are coefficients of the given polynomials and is typically smaller than the matrices obtained by previous approaches. The matrix determinant is a non-trivial multiple of the sparse resultant from which the sparse resultant itself can be recovered. The algorithm is incremental in the sense that successively larger matrices are constructed until one is found with the above properties. For multigraded systems, the new algorithm produces optimal matrices, i.e. expresses the sparse resultant as a single determinant. An implementation of the algorithm is described and experimental results are presented. In addition, we propose an efficient algorithm for computing the mixed volume of n polynomials in n variables. This computation provides an upper bound on the number of common isolated roots. A publicly available implementation of the algorithm is presented and empirical results are reported which suggest that it is the fastest mixed volume code to date.

Matrices in elimination theory

The last decade has witnessed the rebirth of resultant methods as a powerful computational tool for variable elimination and polynomial system solving. In particular, the advent of sparse elimination theory and toric varieties has provided ways to exploit the structure of polynomials encountered in a number of scientific and engineering applications. On the other hand, the Bezoutian reveals itself as an important tool in many areas connected to elimination theory and has its own merits, leading to new developments in effective algebraic geometry. This survey unifies the existing work on resultants, with emphasis on constructing matrices that generalize the classic matrices named after Sylvester, Bezout and Macaulay. The properties of the different matrix formulations are presented, including some complexity issues, with emphasis on variable elimination theory. We compare toric resultant matrices to Macaulay?s matrix and further conjecture the generalization of Macaulay?s exact rational expression for the resultant polynomial to the toric case. A new theorem proves that the maximal minor of a Bezout matrix is a non-trivial multiple of the resultant. We discuss applications to constructing monomial bases of quotient rings and multiplication maps, as well as to system solving by linear algebra operations. Lastly, degeneracy issues, a major preoccupation in practice, are examined. Throughout the presentation, examples are used for illustration and open questions are stated in order to point the way to further research.

Certified approximate univariate GCDs

Abstract We study the approximate GCD of two univariate polynomials given with limited accuracy or, equivalently, the exact GCD of the perturbed polynomials within some prescribed tolerance. A perturbed polynomial is regarded as a family of polynomials in a classification space, which leads to an accurate analysis of the computation. Considering only the Sylvester matrix singular values, as is frequently suggested in the literature, does not suffice to solve the problem completely, even when the extended euclidean algorithm is also used. We provide a counterexample that illustrates this claim and indicates the problems hardness. SVD computations on subresultant matrices lead to upper bounds on the degree of the approximate GCD. Further use of the subresultant matrices singular values yields an approximate syzygy of the given polynomials, which is used to establish a gap theorem on certain singular values that certifies the maximum-degree approximate GCD. This approach leads directly to an algorithm for computing the approximate GCD polynomial. Lastly, we suggest the use of weighted norms in order to sharpen the theorems conditions in a more intrinsic context.

A subdivision-based algorithm for the sparse resultant

Multivariate resultants generalize the Sylvester resultant of two polynomials and characterize the solvability of a polynomial system. They also reduce the computation of all common roots to a problem in linear algebra. We propose a determinantal formula for the sparse resultant of an arbitrary system of n + 1 polynomials in n variables. This resultant generalizes the classical one and has significantly lower degree for polynomials that are sparse in the sense that their mixed volume is lower than their Bézout number. Our algorithm uses a mixed polyhedral subdivision of the Minkowski sum of the Newton polytopes in order to construct a Newton matrix. Its determinant is a nonzero multiple of the sparse resultant and the latter equals the GCD of at most n + 1 such determinants. This construction implies a restricted version of an effective sparse Nullstellensatz. For an arbitrary specialization of the coefficients, there are two methods that use one extra variable and yield the sparse resultant. This is the first algorithm to handle the general case with complexity polynomial in the resultant degree and simply exponential in n. We conjecture its extension to producing an exact rational expression for the sparse resultant.

An Efficient Algorithm for the Sparse Mixed Resultant

We propose a compact formula for the mixed resultant of a system of n+1 sparse Laurent polynomials in n variables. Our approach is conceptually simple and geometric, in that it applies a mixed subdivision to the Minkowski Sum of the input Newton polytopes. It constructs a matrix whose determinant is a non-zero multiple of the resultant so that the latter can be defined as the GCD of n + 1 such determinants. For any specialization of the coefficients there are two methods which use one extra perturbation variable and return the resultant. Our algorithm is the first to present a determinantal formula for arbitrary systems; moreover, its complexity for unmixed systems is polynomial in the resultant degree. Further empirical results suggest that this is the most efficient method to date for sparse elimination.

Computer Algebra Methods for Studying and Computing Molecular Conformations

Abstract. A relatively new branch of computational biology has been emerging as an effort to supplement traditional techniques of large scale search in drug design by structure-based methods, in order to improve efficiency and guarantee completeness. This paper studies the geometric structure of cyclic molecules, in particular the enumeration of all possible conformations, which is crucial in finding the energetically favorable geometries, and the identification of all degenerate conformations. Recent advances in computational algebra are exploited, including distance geometry, sparse polynomial theory, and matrix methods for numerically solving nonlinear multivariate polynomial systems. Moreover, we propose a complete array of computer algebra and symbolic computational geometry methods for modeling the rigidity constraints, formulating the problems in algebraic terms and, lastly, visualizing the computed conformations. The use of computer algebra systems and of public domain software is illustrated, in addition to more specialized programs developed by the authors, which are also freely available. Throughout our discussion, we show the relevance of successful paradigms and algorithms from geometry and robot kinematics to computational biology.

Real Algebraic Numbers: Complexity Analysis and Experimentation

We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ ?, using Sturm (-Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both methods and we give an asymptotic complexity bound of

Symbolic and numeric methods for exploiting structure in constructing resultant matrices

\mathcal{\tilde O}_B(d^4 \tau^2)

On the Complexity of Sparse Elimination

. This matches the best known bounds for binary subdivision solvers. Moreover, we generalize this to cover the non square-free polynomials and show that within the same complexity we can also compute the multiplicities of the roots. We also consider algorithms for sign evaluation, comparison of real algebraic numbers and simultaneous inequalities, and we improve the known bounds at least by a factor of d. Finally, we present our C++ implementation in synaps and some preliminary experiments on various data sets.

A General Approach to Removing Degeneracies

Resultant characterize the existence of roots of systems of multivariate nonlinear polynomial equations, while their matrices reduce the computation of all common zeros to a problem in linear algebra. Sparse elimination theory has introduced the sparse (or toric) resultant, which takes into account the sparse structure of the polynomials. The construction of sparse resultant, or Newton, matrices is the critical step in the computation of the multivariate resultant and the solution of a nonlinear system. We reveal and exploit the quasi-Toeplitz structure of the Newton matrix, thus decreasing the time complexity of constructing such matrices by roughly one order of magnitude to achieve quasi-quadratic complexity in the matrix dimension. The space complexity is also decreased analogously. These results imply similar improvements in the complexity of computing the resultant polynomial itself and of solving zero-dimensional systems. Our approach relies on fast vector-by-matrix multiplication and uses the following two methods as building blocks. First, a fast and numerically stable method for determining the rank of rectangular matrices, which works exclusively over floating point arithmetic. Second, exact polynomial arithmetic algorithms that improve upon the complexity of polynomial multiplication under our model of sparseness, offering bounds linear in the number of variables and the number of non-zero terms.