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Hilbert-Samuel and Hilbert-Kunz functions of zero-dimensional ideals
The Hilbert-Samuel function measures the length of powers of a zero-dimensional ideal in a local ring. Samuel showed that over a local ring these lengths agree with a polynomial, called the Hilbert-Samuel polynomial, for sufficiently large powers of the ideal. We examine the coefficients of this polynomial in the case the ideal is generated by a system of parameters, focusing much of our attention on the second Hilbert coefficient. We also consider the Hilbert-Kunz function, which measures the length of Frobenius powers of an ideal in a ring of positive characteristic. In particular, we examine a conjecture of Watanabe and Yoshida comparing the Hilbert-Kunz multiplicity and the length of the ideal and provide a proof in the graded case. ^
McDonnell, Lori, "Hilbert-Samuel and Hilbert-Kunz functions of zero-dimensional ideals" (2011). ETD collection for University of Nebraska - Lincoln. AAI3450102.