Computer Science

In this paper, we propose two new deterministic interpolation algorithms for a sparse multivariate polynomial given as a standard black-box by introducing new Kronecker type substitutions. Let $f\in \RB[x_1,\dots,x_n]$ be a sparse black-box polynomial with a degree bound D . When $\RB=\C$ or a finite field, our algorithms either have better bit complexity or better bit complexity in D than existing deterministic algorithms. In particular, in the case of deterministic algorithms for standard black-box models, our second algorithm has the current best complexity in D which is the dominant factor in the complexity.

This paper describes an algorithm which computes the characteristic polynomial of a matrix over a field within the same asymptotic complexity, up to constant factors, as the multiplication of two square matrices. Previously, this was only achieved by resorting to genericity assumptions or randomization techniques, while the best known complexity bound with a general deterministic algorithm was obtained by Keller-Gehrig in 1985 and involves logarithmic factors. Our algorithm computes more generally the determinant of a univariate polynomial matrix in reduced form, and relies on new subroutines for transforming shifted reduced matrices into shifted weak Popov matrices, and shifted weak Popov matrices into shifted Popov matrices.

En este trabajo se presenta una propuesta para realizar Diferenciación Automática Anidada utilizando cualquier biblioteca de Diferenciación Automática que permita sobrecarga de operadores. Para calcular las derivadas anidadas en una misma evaluación de la función, la cual se asume que sea anal'itica, se trabaja con el modo forward utilizando una nueva estructura llamada SuperAdouble, que garantiza que se aplique correctamente la diferenciación automática y se calculen el valor y la derivada que se requiera. También se presenta un enfoque algebraico de la Diferenciación Automática y en particular del espacio de los SuperAdoubles. This paper proposes a framework to apply Nested Automatic Differentiation using any library of Automatic Differentiation which allows operator overloading. To compute nested derivatives of a function while it is being evaluated, which is assumed to be analytic, a new structure called SuperAdouble is used in the forward mode. This new class guarantees the correct application of Automatic Differentiation to calculate the value and derivative of a function where is required. Also, an Automatic Differentiation algebraic point of view is presented with particular emphasis in Nested Automatic Differentiation.

Basic principles of set theory have been applied in the context of probability and binary computation. Applying the same principles on inequalities is less common but can be extremely beneficial in a variety of fields. This paper formulates a novel approach to directly apply set operations on inequalities to produce resultant inequalities with differentiable boundaries. The suggested approach uses inequalities of the form Ei: fi(x1,x2,..,xn) and an expression of set operations in terms of Ei like, (E1 and E2) or E3, or can be in any standard form like the Conjunctive Normal Form (CNF) to produce an inequality F(x1,x2,..,xn)<=1 which represents the resulting bounded region from the expressions and has a differentiable boundary. To ensure differentiability of the solution, a trade-off between representation accuracy and curvature at borders (especially corners) is made. A set of parameters is introduced which can be fine-tuned to improve the accuracy of this approach. The various applications of the suggested approach have also been discussed which range from computer graphics to modern machine learning systems to fascinating demonstrations for educational purposes (current use). A python script to parse such expressions is also provided.

We improve certain degree bounds for Grobner bases of polynomial ideals in generic position. We work exclusively in deterministically verifiable and achievable generic positions of a combinatorial nature, namely either strongly stable position or quasi stable position. Furthermore, we exhibit new dimension- (and depth-)dependent upper bounds for the Castelnuovo-Mumford regularity and the degrees of the elements of the reduced Grobner basis (w.r.t. the degree reverse lexicographical ordering) of a homogeneous ideal in these positions.

In this paper, we develop a new approach to the discrimi-nant of a complete intersection curve in the 3-dimensional projective space. By relying on the resultant theory, we first prove a new formula that allows us to define this discrimi-nant without ambiguity and over any commutative ring, in particular in any characteristic. This formula also provides a new method for evaluating and computing this discrimi-nant efficiently, without the need to introduce new variables as with the well-known Cayley trick. Then, we obtain new properties and computational rules such as the covariance and the invariance formulas. Finally, we show that our definition of the discriminant satisfies to the expected geometric property and hence yields an effective smoothness criterion for complete intersection space curves. Actually, we show that in the generic setting, it is the defining equation of the discriminant scheme if the ground ring is assumed to be a unique factorization domain.

The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several variables (ACSV) makes it possible to compute asymptotic expansions. We consider these methods from the point of view of effectivity. In particular, given a rational function, ACSV requires one to determine a (generically) finite collection of points that are called critical and minimal. Criticality is an algebraic condition, meaning it is well treated by classical methods in computer algebra, while minimality is a semi-algebraic condition describing points on the boundary of the domain of convergence of a multivariate power series. We show how to obtain dominant asymptotics for the diagonal coefficient sequence of multivariate rational functions under some genericity assumptions using symbolic-numeric techniques. To our knowledge, this is the first completely automatic treatment and complexity analysis for the asymptotic enumeration of rational functions in an arbitrary number of variables.

Effective matrix methods for solving standard linear algebra problems in a commutative domains are discussed. Two of them are new. There are a methods for computing adjoined matrices and solving system of linear equations in a commutative domains.

Creative telescoping is a powerful computer algebra paradigm -initiated by Doron Zeilberger in the 90's- for dealing with definite integrals and sums with parameters. We address the mixed continuous-discrete case, and focus on the integration of bivariate hypergeometric-hyperexponential terms. We design a new creative telescoping algorithm operating on this class of inputs, based on a Hermite-like reduction procedure. The new algorithm has two nice features: it is efficient and it delivers, for a suitable representation of the input, a minimal-order telescoper. Its analysis reveals tight bounds on the sizes of the telescoper it produces.

Galois field (GF) arithmetic is used to implement critical arithmetic components in communication and security-related hardware, and verification of such components is of prime importance. Current techniques for formally verifying such components are based on computer algebra methods that proved successful in verification of integer arithmetic circuits. However, these methods are sequential in nature and do not offer any parallelism. This paper presents an algebraic functional verification technique of gate-level GF (2m ) multipliers, in which verification is performed in bit-parallel fashion. The method is based on extracting a unique polynomial in Galois field of each output bit independently. We demonstrate that this method is able to verify an n-bit GF multiplier in n threads. Experiments performed on pre- and post-synthesized Mastrovito and Montgomery multipliers show high efficiency up to 571 bits.