Recursive Polynomial Remainder Sequence and its Subresultants

We introduce concepts of "recursive polynomial remainder sequence (PRS)" and "recursive subresultant," along with investigation of their properties. A recursive PRS is defined as, if there exists the GCD (greatest common divisor) of initial polynomials, a sequence of PRSs calculated "recursively" for the GCD and its derivative until a constant is derived, and recursive subresultants are defined by determinants representing the coefficients in recursive PRS as functions of coefficients of initial polynomials. We give three different constructions of subresultant matrices for recursive subresultants; while the first one is built-up just with previously defined matrices thus the size of the matrix increases fast as the recursion deepens, the last one reduces the size of the matrix drastically by the Gaussian elimination on the second one which has a "nested" expression, i.e. a Sylvester matrix whose elements are themselves determinants.Comment: 30 pages. Preliminary versions of this paper have been presented at CASC 2003 (arXiv:0806.0478 [math.AC]) and CASC 2005 (arXiv:0806.0488 [math.AC]

Light types for polynomial time computation in lambda-calculus

We propose a new type system for lambda-calculus ensuring that well-typed programs can be executed in polynomial time: Dual light affine logic (DLAL). DLAL has a simple type language with a linear and an intuitionistic type arrow, and one modality. It corresponds to a fragment of Light affine logic (LAL). We show that contrarily to LAL, DLAL ensures good properties on lambda-terms: subject reduction is satisfied and a well-typed term admits a polynomial bound on the reduction by any strategy. We establish that as LAL, DLAL allows to represent all polytime functions. Finally we give a type inference procedure for propositional DLAL.Comment: 20 pages (including 10 pages of appendix). (revised version; in particular section 5 has been modified). A short version is to appear in the proceedings of the conference LICS 2004 (IEEE Computer Society Press

GPGCD: An iterative method for calculating approximate GCD of univariate polynomials

We present an iterative algorithm for calculating approximate greatest common divisor (GCD) of univariate polynomials with the real or the complex coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. The problem of approximate GCD is transfered to a constrained minimization problem, then solved with the so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. We demonstrate that, in some test cases, our algorithm calculates approximate GCD with perturbations as small as those calculated by a method based on the structured total least norm (STLN) method and the UVGCD method, while our method runs significantly faster than theirs by approximately up to 30 or 10 times, respectively, compared with their implementation. We also show that our algorithm properly handles some ill-conditioned polynomials which have a GCD with small or large leading coefficient.Comment: Preliminary versions have been presented as doi:10.1145/1576702.1576750 and arXiv:1007.183