Faculty Mentor
Jozsef Losonczy
Major/Area of Research
Mathematics
Description
An ideal I in a polynomial ring k[x1,...,xn] is a nonempty set closed under
addition satisfying hf _ I whenever h _ k[x1,...,xn] and f _ I. We say I is finitely
generated if there exist f1,...,fs _ I such that every element of I is of the form
h1f1+ΣΣΣ+hsfs for some h1,...,hs _ k[x1,...,xn]. By a fundamental result called
the Hilbert Basis Theorem, every polynomial ideal is finitely generated. Every
polynomial ideal has a special finite generating set called a Groebner basis.
A Groebner basis {g1,...,gt} is a set of polynomials in I such that the ideal
generated by the leading terms of polynomials in I coincides with the ideal
generated by the leading terms of g1,...,gt. Groebner bases were established
by Bruno Buchberger, and can be computed by the algorithm developed in
his 1965 PhD thesis. Groebner bases have many valuable applications in
commutative algebra. By their nature, they provide a means of ideal description.
They also give a definitive result in determining whether or not a
polynomial is a member of an ideal. Groebner bases are utilized in algebraic
geometry as well, such as in using elimination theory to solve a system of
polynomial equations, and in providing an implicit representation of a parametric
curve or surface. Lastly, Groebner bases can be used to describe the
results of certain operations on ideals, such as the intersection of two ideals
and the radical of an ideal, the latter being fundamental to the relationship
between ideals and varieties.
Included in
Applications of Groebner Bases in Commutative Algebra and Algebraic Geometry
An ideal I in a polynomial ring k[x1,...,xn] is a nonempty set closed under
addition satisfying hf _ I whenever h _ k[x1,...,xn] and f _ I. We say I is finitely
generated if there exist f1,...,fs _ I such that every element of I is of the form
h1f1+ΣΣΣ+hsfs for some h1,...,hs _ k[x1,...,xn]. By a fundamental result called
the Hilbert Basis Theorem, every polynomial ideal is finitely generated. Every
polynomial ideal has a special finite generating set called a Groebner basis.
A Groebner basis {g1,...,gt} is a set of polynomials in I such that the ideal
generated by the leading terms of polynomials in I coincides with the ideal
generated by the leading terms of g1,...,gt. Groebner bases were established
by Bruno Buchberger, and can be computed by the algorithm developed in
his 1965 PhD thesis. Groebner bases have many valuable applications in
commutative algebra. By their nature, they provide a means of ideal description.
They also give a definitive result in determining whether or not a
polynomial is a member of an ideal. Groebner bases are utilized in algebraic
geometry as well, such as in using elimination theory to solve a system of
polynomial equations, and in providing an implicit representation of a parametric
curve or surface. Lastly, Groebner bases can be used to describe the
results of certain operations on ideals, such as the intersection of two ideals
and the radical of an ideal, the latter being fundamental to the relationship
between ideals and varieties.