arXiv subjects

David Sevilla

Publications and source records attributed to David Sevilla.

24 records · Page 2Linked to original sources

arXiv2008

Computation of unirational fields

One of the main contributions which Volker Weispfenning made to mathematics is related to Groebner bases theory. In this paper we present an algorithm for computing all algebraic intermediate subfields in a separably generated unirational field extension (which in particular includes the zero characteristic case). One of the main tools is Groebner bases theory. Our algorithm also requires computing primitive elements and factoring over algebraic extensions. Moreover, the method can be extended to finitely generated K-algebras.

cs.SC↗

arXiv2008

Building counterexamples to generalizations for rational functions of Ritt's decomposition theorem

The classical Ritt's Theorems state several properties of univariate polynomial decomposition. In this paper we present new counterexamples to Ritt's first theorem, which states the equality of length of decomposition chains of a polynomial, in the case of rational functions. Namely, we provide an explicit example of a rational function with coefficients in Q and two decompositions of different length. Another aspect is the use of some techniques that could allow for other counterexamples, namely, relating groups and decompositions and using the fact that the alternating group A_4 has two subgroup chains of different lengths; and we provide more information about the generalizations of another property of polynomial decomposition: the stability of the base field. We also present an algorithm for computing the fixing group of a rational function providing the complexity over Q.

math.AC↗

arXiv2008

Computation of unirational fields (extended abstract)

In this paper we present an algorithm for computing all algebraic intermediate subfields in a separably generated unirational field extension (which in particular includes the zero characteristic case). One of the main tools is Groebner bases theory. Our algorithm also requires computing computing primitive elements and factoring over algebraic extensions. Moreover, the method can be extended to finitely generated K-algebras.

cs.SC↗

arXiv2008

Prediciendo el generador cuadratico (in Spanish)

Let p be a prime and a, c be integers such that a<>0 mod p. The quadratic generator is a sequence (u_n) of pseudorandom numbers defined by u_{n+1}=a*(u_n)^2+c mod p. In this article we probe that if we know sufficiently many of the most significant bits of two consecutive values u_n, u_{n+1}, then we can compute the seed u_0 except for a small number of exceptional values. ----- Sean p un primo, a y c enteros tales que a<>0 mod p. El generador cuadratico es una sucesion (u_n) de numeros pseudoaleatorios definidos por la relacion u_{n+1}=a*(u_n)^2+c mod p. En este trabajo demostramos que si conocemos un numero suficientemente grande de los bits mas significativos para dos valores consecutivos u_n, u_{n+1}, entonces podemos descubrir en tiempo polinomial la semilla u_0, excepto para un conjunto pequeno de valores excepcionales.

cs.CR↗

arXiv2008

On Ritt's decomposition Theorem in the case of finite fields

A classical theorem by Ritt states that all the complete decomposition chains of a univariate polynomial satisfying a certain tameness condition have the same length. In this paper we present our conclusions about the generalization of these theorem in the case of finite coefficient fields when the tameness condition is dropped.

cs.SC↗

arXiv2008

Decomposing replicable functions

We describe an algorithm to decompose rational functions from which we determine the poset of groups fixing these functions.

math.NT↗