BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Michael Curran (Williams College) DTSTART:20200601T160000Z DTEND:20200601T162500Z DTSTAMP:20260423T011219Z UID:CANT2020/71 DESCRIPTION:Title: Ehrhart theory and an explicit version of Khovanskii's theorem\nby M ichael Curran (Williams College) as part of Combinatorial and additive num ber theory (CANT 2021)\n\n\nAbstract\nA remarkable theorem due to Khovansk ii asserts that for any finite subset $A$\nof an abelian group\, the cardi nality of the $h$-fold sumset $hA$ grows like a polynomial\nfor all suffic iently large $h$.\nHowever\, neither the polynomial nor what sufficiently large means are understood in general.\nWe use Ehrhart theory to give a ne w proof of Khovanskii's theorem when\n$A \\subset \\mathbb{Z}^d$ that give s new insights into the growth of the cardinality\nof sumsets. Our approac h allows us to obtain explicit formulae for $|hA|$ whenever\n$A \\subset \ \mathbb{Z}^d$ contains $d + 2$ points that are valid for \\emph{all} $h$.\ nIn the case that the convex hull $\\Delta_A$ of $A$ is a $d$-dimensional simplex\,\nwe can also show that $|hA|$ grows polynomially whenever\n$h \\ geq \\text{vol}(\\Delta_A) \\cdot d! - |A| + 2$.\n LOCATION:https://researchseminars.org/talk/CANT2020/71/ END:VEVENT END:VCALENDAR