Interactive certificate for the verification of Wiedemann's Krylov sequence: application to the certification of the determinant, the minimal and the characteristic polynomials of sparse matrices

Certificates to a linear algebra computation are additional data structures for each output, which can be used by a-possibly randomized- verification algorithm that proves the correctness of each output. Wiede-mann's algorithm projects the Krylov sequence obtained by repeatedly multiplying a vector by a matrix to obtain a linearly recurrent sequence. The minimal polynomial of this sequence divides the minimal polynomial of the matrix. For instance, if the $n\times n$ input matrix is sparse with n 1+o(1) non-zero entries, the computation of the sequence is quadratic in the dimension of the matrix while the computation of the minimal polynomial is n 1+o(1), once that projected Krylov sequence is obtained. In this paper we give algorithms that compute certificates for the Krylov sequence of sparse or structured $n\times n$ matrices over an abstract field, whose Monte Carlo verification complexity can be made essentially linear. As an application this gives certificates for the determinant, the minimal and characteristic polynomials of sparse or structured matrices at the same cost

Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits

We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of for00504925mulas and weakly skew circuits. Our representations produce matrices of much smaller dimensions than those given in the convex geometry literature when applied to polynomials having a concise representation (as a sum of monomials, or more generally as an arithmetic formula or a weakly skew circuit). These representations are valid in any field of characteristic different from 2. In characteristic 2 we are led to an almost complete solution to a question of B\"urgisser on the VNP-completeness of the partial permanent. In particular, we show that the partial permanent cannot be VNP-complete in a finite field of characteristic 2 unless the polynomial hierarchy collapses.Comment: To appear in the AMS Contemporary Mathematics volume on Randomization, Relaxation, and Complexity in Polynomial Equation Solving, edited by Gurvits, Pebay, Rojas and Thompso

Elimination-based certificates for triangular equivalence and rank profiles

International audienceIn this paper, we give novel certificates for triangular equivalence and rank profiles. These certificates enable somebody to verify the row or column rank profiles or the whole rank profile matrix faster than recomputing them, with a negligible overall overhead. We first provide quadratic time and space non-interactive certificates saving the logarithmic factors of previously known ones. Then we propose interactive certificates for the same problems whose Monte Carlo verification complexity requires a small constant number of matrix-vector multiplications, a linear space, and a linear number of extra field operations, with a linear number of interactions. As an application we also give an interactive protocol, certifying the determinant or the signature of dense matrices, faster for the Prover than the best previously known one. Finally we give linear space and constant round certificates for the row or column rank profiles

06271 Abstracts Collection -- Challenges in Symbolic Computation Software

From 02.07.06 to 07.07.06, the Dagstuhl Seminar 06271 ``Challenges in Symbolic Computation Software\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Linear Time Interactive Certificates for the Minimal Polynomial and the Determinant of a Sparse Matrix

International audienceComputational problem certificates are additional data structures for each output, which can be used by a—possibly randomized—verification algorithm that proves the correctness of each output. In this paper, we give an algorithm that computes a certificate for the minimal polynomial of sparse or structured n×n matrices over an abstract field, of sufficiently large cardinality, whose Monte Carlo verification complexity requires a single matrix-vector multiplication and a linear number of extra field operations. We also propose a novel preconditioner that ensures irreducibility of the characteristic polynomial of the generically preconditioned matrix. This preconditioner takes linear time to be applied and uses only two random entries. We then combine these two techniques to give algorithms that compute certificates for the determinant, and thus for the characteristic polynomial, whose Monte Carlo verification complexity is therefore also linear

Effective Hilbert Irreducibility

n this paper we prove by entirely elementary means a very effective version of the Hilbert Irreducibility - n Theorem. We then apply our theorem to construct a probabilistic irreducibility test for sparse multivariate poly omials over arbitrary perfect fields. For the usual coefficient fields the test runs in polynomial time in the input K size. eywords. Hilbert Irreducibility Theorem, Probabilistic Algorithms, Polynomial Factorization, Sparse Polynomials. 1. Introduction s i The question whether a polynomial with coefficients in a unique factorization domain i rreducible poses an old problem. Recently, several new algorithms for univariate and mul- - w tivariate factorization over various coefficient domains have been proposed within the frame ork of polynomial time complexity, see e.g. Berlekamp (1970), Lenstra et al. (1982), Kaltoj fen (1985a), Chistov and Grigoryev (1982), Landau (1985). All algorithms in the references ust given are polynomial in l (n +1) , where l is the num..

Single-Factor Hensel Lifting and its Application to the Straight-Line Complexity of Certain Polynomials

Three theorems are presented that establish polynomial straight-line complexity for certain operations on polynomials given by straight-line programs of unbounded input degree he first theorem shows how to compute a higher order partial derivative in a single variable. l The other two theorems impose the degree of the output polynomial as a parameter of the ength of the output program. First it is shown that if a straight-line program computes an l b arbitrary power of a multivariate polynomial, that polynomial also admits a polynomia ounded straight-line computation. Second, any factor of a multivariate polynomial given by e c a division-free straight-line program with relatively prime co-factor also admits a straight-lin omputation of length polynomial in the input length and the degree of the factor. This result t is based on a new Hensel lifting process, one where only one factor image is lifted back to he original factor. As an application we get that the greatest common divi..