Computer Science

In this paper, we consider the existence of a factorization of a monic, bounded motion polynomial. We prove existence of factorizations, possibly after multiplication with a real polynomial and provide algorithms for computing polynomial factor and factorizations. The first algorithm is conceptually simpler but may require a high degree of the polynomial factor. The second algorithm gives an optimal degree.

Following the works by Lin et al. (Circuits Syst. Signal Process. 20(6): 601-618, 2001) and Liu et al. (Circuits Syst. Signal Process. 30(3): 553-566, 2011), we investigate how to factorize a class of multivariate polynomial matrices. The main theorem in this paper shows that an l×m polynomial matrix admits a factorization with respect to a polynomial if the polynomial and all the (l−1)×(l−1) reduced minors of the matrix generate the unit ideal. This result is a further generalization of previous works, and based on this, we give an algorithm which can be used to factorize more polonomial matrices. In addition, an illustrate example is given to show that our main theorem is non-trivial and valuable.

We compute minimal bases of solutions for a general interpolation problem, which encompasses Hermite-Padé approximation and constrained multivariate interpolation, and has applications in coding theory and security. This problem asks to find univariate polynomial relations between m vectors of size σ ; 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 O ~ ( m ω−1 σ) field operations, where ω is the exponent of matrix multiplication and the notation O ~ (⋅) indicates that logarithmic terms are omitted. Earlier works, in the case of Hermite-Padé approximation and in the general interpolation case, compute non-normalized bases. Since for arbitrary shifts such bases may have size Θ( m 2 σ) , the cost bound O ~ ( m ω−1 σ) 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 O(mσ) . 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.

We give a Las Vegas algorithm which computes the shifted Popov form of an m×m nonsingular polynomial matrix of degree d in expected O ˜ ( m ω d) field operations, where ω is the exponent of matrix multiplication and O ˜ (⋅) indicates that logarithmic factors are omitted. This is the first algorithm in O ˜ ( m ω d) for shifted row reduction with arbitrary shifts. Using partial linearization, we reduce the problem to the case d≤⌈σ/m⌉ where σ is the generic determinant bound, with σ/m bounded from above by both the average row degree and the average column degree of the matrix. The cost above becomes O ˜ ( m ω ⌈σ/m⌉) , improving upon the cost of the fastest previously known algorithm for row reduction, which is deterministic. Our algorithm first builds a system of modular equations whose solution set is the row space of the input matrix, and then finds the basis in shifted Popov form of this set. We give a deterministic algorithm for this second step supporting arbitrary moduli in O ˜ ( m ω−1 σ) field operations, where m is the number of unknowns and σ is the sum of the degrees of the moduli. This extends previous results with the same cost bound in the specific cases of order basis computation and M-Padé approximation, in which the moduli are products of known linear factors.

In 1977, Strassen invented a famous baby-step/giant-step algorithm that computes the factorial N! in arithmetic complexity quasi-linear in N − − √ . In 1988, the Chudnovsky brothers generalized Strassen's algorithm to the computation of the N -th term of any holonomic sequence in essentially the same arithmetic complexity. We design q -analogues of these algorithms. We first extend Strassen's algorithm to the computation of the q -factorial of N , then Chudnovskys' algorithm to the computation of the N -th term of any q -holonomic sequence. Both algorithms work in arithmetic complexity quasi-linear in N − − √ ; surprisingly, they are simpler than their analogues in the holonomic case. We provide a detailed cost analysis, in both arithmetic and bit complexity models. Moreover, we describe various algorithmic consequences, including the acceleration of polynomial and rational solving of linear q -differential equations, and the fast evaluation of large classes of polynomials, including a family recently considered by Nogneng and Schost.

We address the question of computing one selected term of an algebraic power series. In characteristic zero, the best algorithm currently known for computing the N th coefficient of an algebraic series uses differential equations and has arithmetic complexity quasi-linear in N − − √ . We show that over a prime field of positive characteristic p , the complexity can be lowered to O(logN) . The mathematical basis for this dramatic improvement is a classical theorem stating that a formal power series with coefficients in a finite field is algebraic if and only if the sequence of its coefficients can be generated by an automaton. We revisit and enhance two constructive proofs of this result for finite prime fields. The first proof uses Mahler equations, whose sizes appear to be prohibitively large. The second proof relies on diagonals of rational functions; we turn it into an efficient algorithm, of complexity linear in logN and quasi-linear in p .

The row (resp. column) rank profile of a matrix describes the stair-case shape of its row (resp. column) echelon form. We here propose 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 show that this normal form exists and is unique over any ring, provided that the notion of McCoy's rank is used, in the presence of zero divisors. We then explore the conditions for a Gaussian elimination algorithm to compute all or part of this invariant, through the corresponding PLUQ decomposition. This enlarges the set of known Elimination variants that compute row or column rank profiles. As a consequence a new Crout base case variant significantly improves the practical efficiency of previously known implementations over a finite field. With matrices of very small rank, we also generalize the techniques of Storjohann and Yang to the computation of the rank profile matrix, achieving an ( r ω +mn ) 1+o(1) time complexity for an m×n matrix of rank r , where ω is the exponent of matrix multiplication. Finally, by give connections to the Bruhat decomposition, and several of its variants and generalizations. Thus, our algorithmic improvements for the PLUQ factorization, and their implementations, directly apply to these decompositions. In particular, we show how a PLUQ decomposition revealing the rank profile matrix also reveals both a row and a column echelon form of the input matrix or of any of its leading sub-matrices, by a simple post-processing made of row and column permutations.

We give an algorithm for computing all roots of polynomials over a univariate power series ring over an exact field K . More precisely, given a precision d , and a polynomial Q whose coefficients are power series in x , the algorithm computes a representation of all power series f(x) such that Q(f(x))=0mod x d . The algorithm works unconditionally, in particular also with multiple roots, where Newton iteration fails. Our main motivation comes from coding theory where instances of this problem arise and multiple roots must be handled. The cost bound for our algorithm matches the worst-case input and output size d°(Q) , up to logarithmic factors. This improves upon previous algorithms which were quadratic in at least one of d and °(Q) . Our algorithm is a refinement of a divide \& conquer algorithm by Alekhnovich (2005), where the cost of recursive steps is better controlled via the computation of a factor of Q which has a smaller degree while preserving the roots.

The article considers linear functions of many (n) variables - multilinear polynomials (MP). The three-steps evaluation is presented that uses the minimal possible number of floating point operations for non-sparse MP at each step. The minimal number of additions is achieved in the algorithm for fast MP derivatives (FMPD) calculation. The cost of evaluating all first derivatives approaches to only 1/8 of MP evaluation with a growing number of variables. The FMPD algorithm structure exhibits similarity to the Fast Fourier Transformation (FFT) algorithm.

This paper presents new fast algorithms for Hermite interpolation and evaluation over finite fields of characteristic two. The algorithms reduce the Hermite problems to instances of the standard multipoint interpolation and evaluation problems, which are then solved by existing fast algorithms. The reductions are simple to implement and free of multiplications, allowing low overall multiplicative complexities to be obtained. The algorithms are suitable for use in encoding and decoding algorithms for multiplicity codes.