6 research outputs found
Minimal Involutive Bases
In this paper we present an algorithm for construction of minimal involutive
polynomial bases which are Groebner bases of the special form. The most general
involutive algorithms are based on the concept of involutive monomial division
which leads to partition of variables into multiplicative and
non-multiplicative. This partition gives thereby the self-consistent
computational procedure for constructing an involutive basis by performing
non-multiplicative prolongations and multiplicative reductions. Every specific
involutive division generates a particular form of involutive computational
procedure. In addition to three involutive divisions used by Thomas, Janet and
Pommaret for analysis of partial differential equations we define two new ones.
These two divisions, as well as Thomas division, do not depend on the order of
variables. We prove noetherity, continuity and constructivity of the new
divisions that provides correctness and termination of involutive algorithms
for any finite set of input polynomials and any admissible monomial ordering.
We show that, given an admissible monomial ordering, a monic minimal involutive
basis is uniquely defined and thereby can be considered as canonical much like
the reduced Groebner basis.Comment: 22 page
The Theory of Involutive Divisions and an Application to Hilbert Function Computations
Generalising the divisibility relation of terms we introduce the lattice of so-called involutive divisions and define the admissibility of such an involutive division for a given set of terms. Based on this theory we present a new approach for building up a general theory of involutive bases of polynomial ideals. In particular, we give algorithms for checking the involutive basis property and for completing an arbitrary basis to an involutive one. It turns out that our theory is more constructive and more exible than the axiomatic approach to general involutive bases due to Gerdt and Blinkov. Finally, we show that an involutive basis contains more structural information about the ideal of leading terms than a Gröbner basis and that it is straight forward to compute the (affine) Hilbert function of an ideal I from an arbitrary involutive basis of I