33 - On The Existence Of An Arbitrarily Large Number Of Generators For The Presentation Ideal Of a Semigroup Ring.

Abstract

Consider K to be an arbitrary field, and P(n₁,…, n_m) be the ideal of polynomials given by

P(n₁,…, n_m) = {f(x₁, … , x_m) : f(x₁,…,x_m) ∈ K[x₁,…,x_m], f(tⁿ₁, … ,tⁿ_m) = 0, where t is transcendental over K}.

In 1970, J. Herzog showed that the least upper bound on the number of generators of K, for m = 3, is 3. It can be lowered to two, if n₁, n₂, n₃ satisfy a few symmetry conditions. Following that, Bresinsky in 1975, showed that the lowest upper bound on the number of generators of P(n₁, …, n_m), can be arbitrarily large if m is greater than or equal to 4. Recent work by Herzog and Stamate provides a closed form for the number of generators for the semigroup in Bresinsky’s example showing that this number is arbitrarily large but even (precisely 2k, where k is built into Bresinsky’s semigroup and can be any natural number).

Since then a lot of progress has been made in investigating and finding a closed form for the number of generators of the ideal of relations for m greater than or equal to 4. All established work in the field produced examples where this number is always an even number.

However, in 2017, Stamate considers a semigroup suggested by Backelin, which has the following structure.

H = (r(3n+2), r(3n+2)+3, r(3n+2)+3n+1, r(3n+2)+3n+2)

where n is greater than or equal to 2, and r is greater than or equal to 3n+2.

Stamate reports that computations using Singular and GAP indicate that the number of generators for this semigroup is 3n+4, which can be an odd number.

The purpose of this project is to theoretically verify that result. In doing so, the project not only answers a fundamental question in semigroup theory; but also fills the vacuum caused by the lack of any examples with an odd number of generators, thereby completing a 43-year-old question.