Computer Science

Let F q be a finite field. Given two irreducible polynomials f,g over F q , with degf dividing degg , the finite field embedding problem asks to compute an explicit description of a field embedding of F q [X]/f(X) into F q [Y]/g(Y) . When degf=degg , this is also known as the isomorphism problem. This problem, a special instance of polynomial factorization, plays a central role in computer algebra software. We review previous algorithms, due to Lenstra, Allombert, Rains, and Narayanan, and propose improvements and generalizations. Our detailed complexity analysis shows that our newly proposed variants are at least as efficient as previously known algorithms, and in many cases significantly better. We also implement most of the presented algorithms, compare them with the state of the art computer algebra software, and make the code available as open source. Our experiments show that our new variants consistently outperform available software.

We consider the problem of computing univariate polynomial matrices over a field that represent minimal solution bases for a general interpolation problem, some forms of which are the vector M-Padé approximation problem in [Van Barel and Bultheel, Numerical Algorithms 3, 1992] and the rational interpolation problem in [Beckermann and Labahn, SIAM J. Matrix Anal. Appl. 22, 2000]. Particular instances of this problem include the bivariate interpolation steps of Guruswami-Sudan hard-decision and Kötter-Vardy soft-decision decodings of Reed-Solomon codes, the multivariate interpolation step of list-decoding of folded Reed-Solomon codes, and Hermite-Padé approximation. In the mentioned references, the problem is solved using iterative algorithms based on recurrence relations. Here, we discuss a fast, divide-and-conquer version of this recurrence, taking advantage of fast matrix computations over the scalars and over the polynomials. This new algorithm is deterministic, and for computing shifted minimal bases of relations between m vectors of size σ it uses O ( m ω−1 (σ+|s|)) field operations, where ω is the exponent of matrix multiplication, and |s| is the sum of the entries of the input shift s , with min(s)=0 . This complexity bound improves in particular on earlier algorithms in the case of bivariate interpolation for soft decoding, while matching fastest existing algorithms for simultaneous Hermite-Padé approximation.

Let $V\in\mathbb{Q}(i)(\a_1,\dots,\a_n)(\q_1,\q_2)$ be a rationally parametrized planar homogeneous potential of homogeneity degree k≠−2,0,2 . We design an algorithm that computes polynomial \emph{necessary} conditions on the parameters $(\a_1,\dots,\a_n)$ such that the dynamical system associated to the potential V is integrable. These conditions originate from those of the Morales-Ramis-Simó integrability criterion near all Darboux points. The implementation of the algorithm allows to treat applications that were out of reach before, for instance concerning the non-integrability of polynomial potentials up to degree 9 . Another striking application is the first complete proof of the non-integrability of the \emph{collinear three body problem}.

A period of a rational integral is the result of integrating, with respect to one or several variables, a rational function over a closed path. This work focuses particularly on periods depending on a parameter: in this case the period under consideration satisfies a linear differential equation, the Picard-Fuchs equation. I give a reduction algorithm that extends the Griffiths-Dwork reduction and apply it to the computation of Picard-Fuchs equations. The resulting algorithm is elementary and has been successfully applied to problems that were previously out of reach.

Mahler equations relate evaluations of the same function f at iterated b th powers of the variable. They arise in particular in the study of automatic sequences and in the complexity analysis of divide-and-conquer algorithms. Recently, the problem of solving Mahler equations in closed form has occurred in connection with number-theoretic questions. A difficulty in the manipulation of Mahler equations is the exponential blow-up of degrees when applying a Mahler operator to a polynomial. In this work, we present algorithms for solving linear Mahler equations for series, polynomials, and rational functions, and get polynomial-time complexity under a mild assumption. Incidentally, we develop an algorithm for computing the gcrd of a family of linear Mahler operators.

The W-characteristic set of a polynomial ideal is the minimal triangular set contained in the reduced lexicographical Groebner basis of the ideal. A pair (G,C) of polynomial sets is a strong regular characteristic pair if G is a reduced lexicographical Groebner basis, C is the W-characteristic set of the ideal <G>, the saturated ideal sat(C) of C is equal to <G>, and C is regular. In this paper, we show that for any polynomial ideal I with given generators one can either detect that I is unit, or construct a strong regular characteristic pair (G,C) by computing Groebner bases such that I ⊆ sat(C)=<G> and sat(C) divides I, so the ideal I can be split into the saturated ideal sat(C) and the quotient ideal I:sat(C). Based on this strategy of splitting by means of quotient and with Groebner basis and ideal computations, we devise a simple algorithm to decompose an arbitrary polynomial set F into finitely many strong regular characteristic pairs, from which two representations for the zeros of F are obtained: one in terms of strong regular Groebner bases and the other in terms of regular triangular sets. We present some properties about strong regular characteristic pairs and characteristic decomposition and illustrate the proposed algorithm and its performance by examples and experimental results.

We consider the computation of syzygies of multivariate polynomials in a finite-dimensional setting: for a K[ X 1 ,…, X r ] -module M of finite dimension D as a K -vector space, and given elements f 1 ,…, f m in M , the problem is to compute syzygies between the f i 's, that is, polynomials ( p 1 ,…, p m ) in K[ X 1 ,…, X r ] m such that p 1 f 1 +⋯+ p m f m =0 in M . Assuming that the multiplication matrices of the r variables with respect to some basis of M are known, we give an algorithm which computes the reduced Gröbner basis of the module of these syzygies, for any monomial order, using O(m D ω−1 +r D ω log(D)) operations in the base field K , where ω is the exponent of matrix multiplication. Furthermore, assuming that M is itself given as M=K[ X 1 ,…, X r ] n /N , under some assumptions on N we show that these multiplication matrices can be computed from a Gröbner basis of N within the same complexity bound. In particular, taking n=1 , m=1 and f 1 =1 in M , this yields a change of monomial order algorithm along the lines of the FGLM algorithm with a complexity bound which is sub-cubic in D .

Motivated by finding analogues of elliptic curve point counting techniques, we introduce one deterministic and two new Monte Carlo randomized algorithms to compute the characteristic polynomial of a finite rank-two Drinfeld module. We compare their asymptotic complexity to that of previous algorithms given by Gekeler, Narayanan and Garai-Papikian and discuss their practical behavior. In particular, we find that all three approaches represent either an improvement in complexity or an expansion of the parameter space over which the algorithm may be applied. Some experimental results are also presented.

We describe how the extension of a solver for linear differential equations by Kovacic's algorithm helps to improve a method to compute the inverse Mellin transform of holonomic sequences. The method is implemented in the computer algebra package HarmonicSums.

The row (resp. column) rank profile of a matrix describes the staircase shape of its row (resp. column) echelon form. In an ISSAC'13 paper, we proposed a recursive Gaussian elimination that can compute simultaneously the row and column rank profiles of a matrix as well as those of all of its leading sub-matrices, in the same time as state of the art Gaussian elimination algorithms. Here we first study the conditions making a Gaus-sian elimination algorithm reveal this information. Therefore, we propose the definition of a new matrix invariant, the rank profile matrix, summarizing all information on the row and column rank profiles of all the leading sub-matrices. We also explore the conditions for a Gaussian elimination algorithm to compute all or part of this invariant, through the corresponding PLUQ decomposition. As a consequence, we show that the classical iterative CUP decomposition algorithm can actually be adapted to compute the rank profile matrix. Used, in a Crout variant, as a base-case to our ISSAC'13 implementation, it delivers a significant improvement in efficiency. Second, the row (resp. column) echelon form of a matrix are usually computed via different dedicated triangular decompositions. We show here that, from some PLUQ decompositions, it is possible to recover the row and column echelon forms of a matrix and of any of its leading sub-matrices thanks to an elementary post-processing algorithm.