A Sagbi Basis for Some Subalgebras of Polynomial Rings
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Date
2019-03-04
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Addis Ababa University
Abstract
A SAGBI BASIS FOR SOME SUBALGEBRAS OF
POLYNOMIAL RINGS
Dawit Solomon Tadesse
Addis Ababa University, 2018
The term \SAGBI" is an acronym for \Subalgebra Analogue to Gr obner Bases
for Ideals". There exist _nitely generated subalgebras of |[x1; x2; :::; xn] which
have no _nite SAGBI basis with respect to any monomial order. There are also
subalgebras which may or may not have _nite SAGBI basis depending on the
monomial order de_ned on |[x1; x2; :::; xn]. We know that, there are uncountably
many monomial orders de_ned on |[x1; x2; :::; xn] for n _ 2. It is still an important
open problem to classify subalgebras that have a _nite SAGBI basis. In addition
to this, two or more generators of a subalgebra may not form a SAGBI basis.
Torstensson et al. provide su_cient and necessary conditions for two generators
form a SAGBI basis, in the case of univariate polynomial ring. An other important
open problem is to provide a su_cient andnor necessary condition when three or
more generators form a SAGBI basis in the univariate polynomial ring as well
as two or more generators form a SAGBI basis in a multivariate polynomial ring.
This thesis provides su_cient conditions when two generators of a subalgebra form
a SAGBI basis in a multivariate polynomial ring. We also investigate su_cient
and necessary condition when generators with consecutive degrees form a SAGBI
basis in the univariate polynomial ring. Finally we conjecture that subalgebras
generated by two polynomials have a _nite SAGBI basis with respect to any
monomial order. We prove our conjecture in certain cases.
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Keywords
Preliminaries, Monomial Orders, Initial Algebra, Subduction