Ludvig Nyman: Kernels of maps between subsets of quotient rings

Abstract.

This paper explores polynomial quotient rings with monomial ideals. Special interest is taken in the kernels for maps between sets of degree \(d\) and \(d+1\) where \(d\) maximises the value of the Hilbert function. The two classes of ideals that are covered in detail are those on the form \(I = \langle x_1^d, x_2^d, x_3^d, x_1^{d/2} x_2^{d/2} \rangle\) for \(d = 2 + 6n\) as well as \(I = \langle x_1^2, x_2^2, \dots, x_n^2 \rangle\) with a new result regarding the vector basis of the kernel for the latter. Minor results regarding a formula for the Hilbert function and it’s maximum value for the first class of ideals are also discussed and proven.