46 research outputs found
Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix
Given a nonsingular matrix of univariate polynomials over a
field , we give fast and deterministic algorithms to compute its
determinant and its Hermite normal form. Our algorithms use
operations in ,
where is bounded from above by both the average of the degrees of the rows
and that of the columns of the matrix and is the exponent of matrix
multiplication. The soft- notation indicates that logarithmic factors in the
big- are omitted while the ceiling function indicates that the cost is
when . Our algorithms are based
on a fast and deterministic triangularization method for computing the diagonal
entries of the Hermite form of a nonsingular matrix.Comment: 34 pages, 3 algorithm
Computing syzygies in finite dimension using fast linear algebra
We consider the computation of syzygies of multivariate polynomials in afinite-dimensional setting: for a -module of finite dimension as a -vector space, andgiven elements in , the problem is to computesyzygies between the 's, that is, polynomials in such that in. Assuming that the multiplication matrices of the variables with respect to some basis of are known, we give analgorithm which computes the reduced Gr\"obner basis of the module of thesesyzygies, for any monomial order, using operations in the base field , where is theexponent of matrix multiplication. Furthermore, assuming that is itself given as ,under some assumptions on we show that these multiplicationmatrices can be computed from a Gr\"obner basis of within thesame complexity bound. In particular, taking , and in, this yields a change of monomial order algorithm along thelines of the FGLM algorithm with a complexity bound which is sub-cubic in
Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts
We compute minimal bases of solutions for a general interpolation problem,
which encompasses Hermite-Pad\'e approximation and constrained multivariate
interpolation, and has applications in coding theory and security.
This problem asks to find univariate polynomial relations between vectors
of size ; these relations should have small degree with respect to an
input degree shift. For an arbitrary shift, we propose an algorithm for the
computation of an interpolation basis in shifted Popov normal form with a cost
of field operations, where
is the exponent of matrix multiplication and the notation
indicates that logarithmic terms are omitted.
Earlier works, in the case of Hermite-Pad\'e approximation and in the general
interpolation case, compute non-normalized bases. Since for arbitrary shifts
such bases may have size , the cost bound
was feasible only with restrictive
assumptions on the shift that ensure small output sizes. The question of
handling arbitrary shifts with the same complexity bound was left open.
To obtain the target cost for any shift, we strengthen the properties of the
output bases, and of those obtained during the course of the algorithm: all the
bases are computed in shifted Popov form, whose size is always . Then, we design a divide-and-conquer scheme. We recursively reduce
the initial interpolation problem to sub-problems with more convenient shifts
by first computing information on the degrees of the intermediate bases.Comment: 8 pages, sig-alternate class, 4 figures (problems and algorithms
Two-Point Codes for the Generalized GK curve
We improve previously known lower bounds for the minimum distance of certain
two-point AG codes constructed using a Generalized Giulietti-Korchmaros curve
(GGK). Castellanos and Tizziotti recently described such bounds for two-point
codes coming from the Giulietti-Korchmaros curve (GK). Our results completely
cover and in many cases improve on their results, using different techniques,
while also supporting any GGK curve. Our method builds on the order bound for
AG codes: to enable this, we study certain Weierstrass semigroups. This allows
an efficient algorithm for computing our improved bounds. We find several new
improvements upon the MinT minimum distance tables.Comment: 13 page
Computing minimal interpolation bases
International audienceWe consider the problem of computing univariate polynomial matrices over afield that represent minimal solution bases for a general interpolationproblem, some forms of which are the vector M-Pad\'e approximation problem in[Van Barel and Bultheel, Numerical Algorithms 3, 1992] and the rationalinterpolation problem in [Beckermann and Labahn, SIAM J. Matrix Anal. Appl. 22,2000]. Particular instances of this problem include the bivariate interpolationsteps of Guruswami-Sudan hard-decision and K\"otter-Vardy soft-decisiondecodings of Reed-Solomon codes, the multivariate interpolation step oflist-decoding of folded Reed-Solomon codes, and Hermite-Pad\'e approximation. In the mentioned references, the problem is solved using iterative algorithmsbased on recurrence relations. Here, we discuss a fast, divide-and-conquerversion of this recurrence, taking advantage of fast matrix computations overthe scalars and over the polynomials. This new algorithm is deterministic, andfor computing shifted minimal bases of relations between vectors of size it uses field operations, where is the exponent of matrix multiplication, and is the sum of theentries of the input shift , with . This complexity boundimproves in particular on earlier algorithms in the case of bivariateinterpolation for soft decoding, while matching fastest existing algorithms forsimultaneous Hermite-Pad\'e approximation
Beating binary powering for polynomial matrices
The th power of a polynomial matrix of fixed size and degree can be computed by binary powering as fast as multiplying two polynomials of linear degree in . When Fast Fourier Transform (FFT) is available, the resulting arithmetic complexity is \emph{softly linear} in , i.e. linear in with extra logarithmic factors. We show that it is possible to beat binary powering, by an algorithm whose complexity is \emph{purely linear} in , even in absence of FFT. The key result making this improvement possible is that the entries of the th power of a polynomial matrix satisfy linear differential equations with polynomial coefficients whose orders and degrees are independent of . Similar algorithms are proposed for two related problems: computing the th term of a C-recursive sequence of polynomials, and modular exponentiation to the power for bivariate polynomials