On Lengths of $\mathbb{F}_2[x,y,z]/(x^{d_1}, y^{d_2},z^{d_3}, x+y+z)$

Fiona Han; Jennifer Kenkel; Daniel Li; Sridhar Venkatesh; Ashley Wiles

paper

On Lengths of $F_{2} [x, y, z] / (x^{d_{1}}, y^{d_{2}}, z^{d_{3}}, x + y + z)$

arXiv:2306.16574

Abstract

In this paper, we provide a formula for the vector space dimension of the ring $F_{2} [x, y, z] / (x^{d_{1}}, y^{d_{2}}, z^{d_{3}}, x + y + z)$ over $F_{2}$ when $d_{1}, d_{2}, d_{3}$ all lie between successive powers of $2$ . For general $d_{1}, d_{2}, d_{3}$ , we provide a simple algorithm to calculate the vector space dimension of $F_{2} [x, y, z] / (x^{d_{1}}, y^{d_{2}}, z^{d_{3}}, x + y + z)$ by combining our formula with certain results of Chungsim Han (1992).

We edited the paper to reflect the fact that Theorem 1.2 of this paper follows from Theorem 3.8 in Han's paper: "The Hilbert-Kunz function of a diagonal hypersurface." We are grateful to Cheng Meng for bringing this to our attention

On Lengths of F2​[x,y,z]/(xd1​,yd2​,zd3​,x+y+z)

Abstract

On Lengths of $F_{2} [x, y, z] / (x^{d_{1}}, y^{d_{2}}, z^{d_{3}}, x + y + z)$