The positivity of the first coefficients of normal Hilbert polynomials

Abstract:

Let $R$ be an analytically unramified local ring with maximal ideal $\mathfrak m$ and $d = \dim R > 0$. If $R$ is unmixed, then $\overline {\mathrm {e}}^{1}_I(R) \geq 0$ for every $\mathfrak m$-primary ideal $I$ in $R$, where $\overline {\mathrm {e}}_I^1(R)$ denotes the first coefficient of the normal Hilbert polynomial of $R$ with respect to $I$. Thus the positivity conjecture on $\overline {\mathrm {e}}_I^1(R)$ posed by Wolmer V. Vasconcelos is settled affirmatively.