Abstract:
If \$P\$ is a lattice polytope (i.e., \$P\$ is the convex hull of finitely many integer points in \${\bf R}^d\$), Ehrhart's theorem asserts that the integer-point counting function \$L_P(m) = \#(mP \cap {\bf Z}^d)\$ is a polynomial in the integer variable \$m\$. Our goal is to study structural properties of Ehrhart polynomials—essentially asking variants of the (way too hard) question which polynomials are Ehrhart polynomials? Similar to the situations with other combinatorial polynomials, it is useful to express \$L_P(m)\$ in different bases. E.g., a theorem of Stanley (1980) says that \$L_P(m)\$, expressed in the polynomial basis \$\binom m d, \binom{m+1} d, \dots, \binom{m+d} d\$, has nonnegative coefficients; these coefficients form the \$h^*\$-vector of \$P\$. More recent work of Breuer (2012) suggests that one ought to also study \$L_P(m)\$ as expressed in the polynomial basis \$\binom{m-1} 0, \binom{m-1}1, \binom{m-1} 2, \dots\$; the coefficients in this basis form the \$f^*\$-vector of \$P\$. We will survey some old and new results (the latter joint work with Danai Deligeorgaki, Max Hlavaczek, and Jéronimo Valencia) about \$f^*\$- and \$h^*\$-vectors, including analogues and dissimilarities with \$f\$- and \$h\$-vectors of polytopes and polyhedral complexes.
Note: This talk begins with a pre-seminar (aimed at graduate students) at 3:30–4:00. The main talk starts at 4:10.
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Meeting ID: 915 4733 5974