Regularity of lex-segment ideals: Some closed formulas and applications

A closed formula for computing the regularity of the lex-segment ideal in terms of the Hilbert function is given. This regularity bounds the one of any ideal with the same Hilbert function. As a consequence, we give explicit expressions to bound the regularity of a projective scheme in terms of the coefficients of the Hilbert polynomial.

We also characterize, in terms of their coefficients, which polynomials are Hilbert polynomials of some projective scheme.

Finally, we provide some applications to estimates for the maximal degree of generators of Gröbner bases in terms of the degrees of defining equations.