37 research outputs found
Deterministically Computing Reduction Numbers of Polynomial Ideals
We discuss the problem of determining reduction number of a polynomial ideal
I in n variables. We present two algorithms based on parametric computations.
The first one determines the absolute reduction number of I and requires
computation in a polynomial ring with (n-dim(I))dim(I) parameters and n-dim(I)
variables. The second one computes via a Grobner system the set of all
reduction numbers of the ideal I and thus in particular also its big reduction
number. However,it requires computations in a ring with n.dim(I) parameters and
n variables.Comment: This new version replaces the earlier version arXiv:1404.1721 and it
has been accepted for publication in the proceedings of CASC 2014, Warsaw,
Polna
A Gröbner-Basis Theory for Divide-and-Conquer Recurrences
International audienceWe introduce a variety of noncommutative polynomials that represent divide-and-conquer recurrence systems. Our setting involves at the same time variables that behave like words in purely noncom-mutative algebras and variables governed by commutation rules like in skew polynomial rings. We then develop a Gröbner-basis theory for left ideals of such polynomials. Strikingly, the nature of commutations generally prevents the leading monomial of a polynomial product to be the product of the leading monomials. To overcome the difficulty, we consider a specific monomial ordering, together with a restriction to monic divisors in intermediate steps. After obtaining an analogue of Buchberger's algorithm, we develop a variant of the 4 algorithm, whose speed we compare
On Computing Groebner Basis in the Rings of Differential Operators
Insa and Pauer presented a basic theory of Groebner basis for differential
operators with coefficients in a commutative ring in 1998, and a criterion was
proposed to determine if a set of differential operators is a Groebner basis.
In this paper, we will give a new criterion such that Insa and Pauer's
criterion could be concluded as a special case and one could compute the
Groebner basis more efficiently by this new criterion
Towards Toric Absolute Factorization
This article gives an algorithm to recover the absolute factorization of a bivariate polynomial, taking into account the geometry of its monomials. It is based on algebraic criterions inherited from algebraic interpolation and toric geometry
High-quality construction of analysis-suitable trivariate NURBS solids by reparameterization methods
High-quality construction of analysis-suitable trivariate NURBS solids by reparameterization method