Research on the Hilbert functions in commutative algebra

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The purpose of this research is to explore the behavior of the Hilbert functions of m-primary ideals in a commutative Noetherian local ring A with the maximal ideal m. The representative of this research gave the following results.1.Research on the first Hilbert coefficients of m-primary idealsThe purpose of this research is to study the equality on the first Hilbert coefficients of m-primary ideals given in Elias and Valla, without assuming that the base local ring(A, m) is Cohen-Macaulay. It is related to the Buchsbaumness of the associated graded ring of m-primary ideals.2.Research on the Hilbert coefficients of parameter idealsThis research shows that the Hilbert coefficients of parameter ideals have uniform bounds if and only if the base local ring A has finitely generated local cohomology modules. The uniform bounds are huge ; the sharp bound for the second Hilbert coefficient in the case where the base local ring has finitely generated local cohomology modules.