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Presented By: Commutative Algebra Seminar - Department of Mathematics

Commutative Algebra Seminar: On lengths of $\mathbb{F}_2[x,y,z]/(x^{d_1}, y^{d_2},z^{d_3}, x+y+z)$

Sridhar Venkatesh (UMich)

While the vector space dimension of the ring $k[x_1, \dots, x_n]/(x_1^{d_1}, \dots, x_n^{d_n})$ can easily be calculated as $\prod_{i=1}^n d_i$, adding in an additional ideal generator $x_1+\dots+x_n$ greatly complicates the problem. In this talk, I will present a formula for the vector space dimension of $\mathbb{F}_2[x,y,z]/(x^{d_1}, y^{d_2},z^{d_3}, x+y+z)$ when $d_1,d_2,d_3$ all lie between successive powers of $2$. Combining this with results of Chungsim Han, we get a complete description of the vector space dimension of $\mathbb{F}_2[x,y,z]/(x^{d_1}, y^{d_2},z^{d_3}, x+y+z)$ for any $d_1,d_2,d_3$. This is joint work with Fiona Han, Jenny Kenkel, Daniel Li, and Ashley Wiles (in fact, this came out of a Winter 2023 LoG(M) project done by Fiona, Daniel and Ashley, where Jenny and I were the mentors).

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