Monoidring/Torisch und normal/Differentielle Signatur/Fakt/en

Let ${}C$ be a pointed rational cone in ${}\mathbb {R} ^{d}$ defining the monoid ${}M=\mathbb {Z} ^{d}\cap C$. Let ${}F_{1},\ldots ,F_{r}$ be the facets of ${}C$ and let ${}\ell _{1},\ldots ,\ell _{r}$ be the describing integral linear forms, meaning that they are positive in the interior of the cone and that they vanish on the corresponding facets. Suppose that ${}K$ is a field of characteristic ${}0$.

Then the differential signature of the monoid ring ${}K[M]$ is

$d!\operatorname {vol} {\left\{P\in C\mid (\ell _{1}+\cdots +\ell _{r})(P)\leq 1\right\}}.$