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

TL;DR

This paper derives a formula and algorithm for calculating the dimension of a specific quotient ring over _2, depending on the exponents, with applications in algebraic structures and computational algebra.

Contribution

It provides a closed-form formula and a practical algorithm for the dimension of a quotient ring involving three variables and a linear relation over _2.

Findings

Derived a formula for the dimension when exponents are between powers of 2

Developed an algorithm for general exponents using the formula and existing results

Applicable to algebraic computations involving polynomial quotient rings over _2

Abstract

In this paper, we provide a formula for the vector space dimension of the ring $\mathbb{F}_2[x,y,z]/(x^{d_1}, y^{d_2},z^{d_3}, x+y+z)$ over $\mathbb{F}_2$ 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 $\mathbb{F}_2[x,y,z]/(x^{d_1}, y^{d_2},z^{d_3}, x+y+z)$ by combining our formula with certain results of Chungsim Han (1992).