## Abstract

Consider K to be an arbitrary field, and P(n_{1},…, n_{m}) be the ideal of polynomials given by

P(n_{1},…, n_{m}) = {f(x_{1}, … , x_{m}) : f(x_{1},…,x_{m}) ∈ K[x_{1},…,x_{m}], f(t^{n}_{1}, … ,t^{n}_{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 n_{1}, n_{2}, n_{3} satisfy a few symmetry conditions. Following that, Bresinsky in 1975, showed that the lowest upper bound on the number of generators of P(n_{1}, …, 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.

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## Metadata

- Event location
Nesbitt 3110

- Event date
3 November 2018

- Date submitted
19 July 2022