In the present paper, we characterize all possible Hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to d, where d is a positive integer. In addition, we prove the following result which is a generalization of Bigatti, Hulett and Pardue's result: Let p ≥ 0 and d > 0 be integers. If the base field is a field of characteristic 0 and there is a graded ideal I whose projective dimension proj dim (I) is smaller than or equal to p and whose regularity reg (I) is smaller than or equal to d, then there exists a monomial ideal L having the maximal graded Betti numbers among graded ideals J which have the same Hilbert function as I and which satisfy proj dim (J) ≤ p and reg (J) ≤ d. We also prove the same fact for squarefree monomial ideals. The main methods for proofs are generic initial ideals and combinatorics on strongly stable ideals.
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