A Graph Theoretic Approach to Regularity of Toric Ideals

Abstract: One measure of the complexity of an ideal is its regularity, which may be difficult to compute with algebraic tools. To study the regularity of certain toric ideals (prime ideals generated by differences of monomials) with quadratic initial ideals, we instead make use of correspondences between the initial ideals and graphs. Moreover, a key theorem by Khosh-Ahang and Moradi gives a method of computing the regularity of a monomial ideal using only the induced matching number of a particular graph associated to it. We find that we can apply this theorem when these graphs are what we call down-left graphs. This research was conducted as part of the 2021 Hobart and William Smith Colleges Mathematics REU, supported by the National Science Foundation under grant no. DMS 1757616.

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