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

Science Diplomacy: Introduction to a Boundary Problem

Scientific advancements, their application through technological development, and world politics have been long acknowledged as affecting each other, and are today more than ever at the heart of global policy. Speaking of ‘science diplomacy’ as the encounter of world politics and the world of science at the heart of these advancements might be a unique window into our time. This potential is what prompts this special issue to gather views from a variety of scholarly and practical viewpoints, linking the well‐established world of reflective practitioners in science diplomacy to the growing field of international relations (IR) scholars theorising this realm. Can speaking of ‘science diplomacy’ situate our attention at the crossroads of science and international relations, and spur greater appreciation for their intersections? This introduction to the special issue summarises the rise of science diplomacy as field of inquiry, and casts questions as to the need to advance, where not reform, these conceptualisations. It defines science diplomacy as a ‘boundary problem’ par excellence and emphasises its ‘productive tension’ that emerges between the various ways of knowing of actors belonging to ‘different social worlds’, seeking to gather a productive tension of views on this theme in the issue

GPGCD: An iterative method for calculating approximate GCD of univariate polynomials

We present an iterative algorithm for calculating approximate greatest common divisor (GCD) of univariate polynomials with the real or the complex coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. The problem of approximate GCD is transfered to a constrained minimization problem, then solved with the so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. We demonstrate that, in some test cases, our algorithm calculates approximate GCD with perturbations as small as those calculated by a method based on the structured total least norm (STLN) method and the UVGCD method, while our method runs significantly faster than theirs by approximately up to 30 or 10 times, respectively, compared with their implementation. We also show that our algorithm properly handles some ill-conditioned polynomials which have a GCD with small or large leading coefficient.Comment: Preliminary versions have been presented as doi:10.1145/1576702.1576750 and arXiv:1007.183

Fast algorithms for computing isogenies between ordinary elliptic curves in small characteristic

The problem of computing an explicit isogeny between two given elliptic curves over F_q, originally motivated by point counting, has recently awaken new interest in the cryptology community thanks to the works of Teske and Rostovstev & Stolbunov. While the large characteristic case is well understood, only suboptimal algorithms are known in small characteristic; they are due to Couveignes, Lercier, Lercier & Joux and Lercier & Sirvent. In this paper we discuss the differences between them and run some comparative experiments. We also present the first complete implementation of Couveignes' second algorithm and present improvements that make it the algorithm having the best asymptotic complexity in the degree of the isogeny.Comment: 21 pages, 6 figures, 1 table. Submitted to J. Number Theor

Hierarchy of general invariants for bivariate LPDOs

We study invariants under gauge transformations of linear partial differential operators on two variables. Using results of BK-factorization, we construct hierarchy of general invariants for operators of an arbitrary order. Properties of general invariants are studied and some examples are presented. We also show that classical Laplace invariants correspond to some particular cases of general invariants.Comment: to appear in J. "Theor.Math.Phys." in May 200