Thom Mulders

Diophantine linear system solving

A sinq)lo randoruixed algorithnl is given for fillding an integer solubion to a s:-stclu of linear Di01)lmnt,ine equations. Givcu as input a s?;stcrrl which admits an int.cger solul.ion, the idgorithru can 1~ used t,o find such a. solution wit,h probabilit~~ ilt least l/2. The running time (nunlber of bit operations) is esscntiall~ cubic in the dimemion of the s?;stem. The malogous result is prcsentctl for 1inca.r spst.cnls over the ring of polqonlials wit.li coefficients from a field.

On Short Multiplications and Divisions

Abstract. Computing only the low degree terms of the product of two univariate polynomials is called a short multiplication. By decomposition into subproblems, a short multiplication can be reduced to appropriate addition of the results of a number of full multiplications. In this paper a new way of choosing the size of the subproblems is proposed. Computing the quotient of two polynomials is called a short division. The ideas used in the short multiplication algorithm are transferred to an algorithm for short divisions. Finally, several applications of short multiplications and divisions are pointed out.

Certified dense linear system solving

Abstract A randomized algorithm is given for solving a system of linear equations over a principal ideal domain. The algorithm returns a solution vector which has minimal denominator. A certificate of minimality is also computed. A given system has a Diophantine solution precisely when the minimal denominator is one. Cost estimates are given for systems over the ring of integers and ring of polynomials with coefficients from a field.

Fast Algorithms for for Linear Algebra Modulo N

Many linear algebra problems over the ring ZN of integers modulo N can be solved by transforming via elementary row operations an n × m input matrix A to Howell form H. The nonzero rows of H give a canonical set of generators for the submodule of (ZN)m generated by the rows of A. In this paper we present an algorithm to recover H together with an invertible transformation matrix P which satisfies PA = H. The cost of the algorithm is O(nmω-1) operations with integers bounded in magnitude by N. This leads directly to fast algorithms for tasks involving ZN-modules, including an O(nmω-1) algorithm for computing the general solution over ZN of the system of linear equations xA = b, where b ∈ (ZN)m.

Rational solutions of singular linear systems

A deterministic algorithm is presented for computing a particular solution to a linear system of equations with polynomial coefficients. Given an <italic>A</italic> ∈ <italic>F</italic>[<italic>x</italic>]<supscrpt><italic>n</italic> × <italic>m</italic></supscrpt> and <italic>b</italic> ∈ <italic>F</italic>[<italic>x</italic>]<supscrpt><italic>n</italic></supscrpt>, where <italic>F</italic> is a field, the algorithm will either return a particular solution <italic>v</italic> ∈ <italic>F</italic>(<italic>x</italic>)<supscrpt><italic>m</italic></supscrpt> to the system <italic>Av</italic> = <italic>b</italic> or determine that the system is inconsistent. The cost of the algorithm is <italic>O</italic>((<italic>n</italic> + <italic>m</italic>)<italic>r</italic><supscrpt>2</supscrpt><italic>d</italic><supscrpt>1 + <italic>ε</italic></supscrpt>) field operations from <italic>F</italic>, where <italic>r</italic> is the rank of <italic>A</italic> and <italic>d</italic> - 1 is a bound for the degrees of entries in <italic>A</italic> and <italic>b</italic>.

On symmetric powers of differential operators

We present alternative algorithms for computing symmetric powers of linear ordinary differential operators. Our algorithms are applicable to operators with coefficients in arbitrary integral domains and become faster than the traditional methods for symmetric powers of sufficiently large order, or over sufficiently complicated coefficient domains. The basic ideaa are also applicable to other computations involving cyclic vector techniques, such as exterior powers of differential or difference operators.

A Note on Subresultants and the Lazard/Rioboo/Trager Formula in Rational Function Integration

An ambiguity in a formula of Lazard, Rioboo and Trager, connecting subresultants and rational function integration, is indicated and examples of incorrect interpretations are given.

A Generalized Sylvester Identity and Fraction-free Random Gaussian Elimination

Sylvester?s identity is a well-known identity that can be used to prove that certain Gaussian elimination algorithms are fraction free. In this paper we will generalize Sylvester?s identity and use it to prove that certain random Gaussian elimination algorithms are fraction free. This can be used to yield fraction free algorithms for solving Ax=b(x? 0) and for the simplex method in linear programming.

The modulo N extended GCD problem for polynomials

We study the following problem: Given a; b;N 2 F [x] with gcd(a; b;N) = 1 and N nonzero, compute a minimal degree f 2 F [x] which satis es gcd(a + fb; N) = 1. We give a deterministic algorithm for solving this problem that is applicable over any eld. The algorithm is designed to solve e ciently a succession of such problems for a xed N . When q = #F > degN the solution will satisfy deg f = 0. When q degN we conjecture that the solution satis es deg f dlogq degNe; in this case the complexity bound we give for the algorithm depends on this conjecture. As an application we demonstrate a deterministic algorithm for computing transforming matrices for the Smith normal form of a nonsingular A 2 F [x] . When q is too small most previous algorithms require working over an algebraic extension of F and may not produce transforming matrices over F [x]. The algorithm we propose will produce transforming matrices over F [x], for elds F of any size.

Certified sparse linear system solving

Abstract A Las Vegas randomized algorithm for solving sparse linear systems over principal ideal domains is described. The algorithm returns a minimal-denominator solution accompanied by a certificate for its minimality or, if no solution exists, a certificate for the inconsistency of the system. The algorithm works for domains of any size, without need of ring extensions.