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

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).

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

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

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

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

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

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

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