Newton polytopes and algebraic hypergeometric series

Alan Adolphson, Steven Sperber, Nicholas M. Katz

Research output: Contribution to journalArticlepeer-review


Let X be the family of hypersurfaces in the odd-dimensional torus T2n+1 defined by a Laurent polynomial f with fixed exponents and variable coefficients. We show that if nΔ, the dilation of the Newton polytope Δ of f by the factor n, contains no interior lattice points, then the Picard-Fuchs equation of W2nHDR2n (X) has a full set of algebraic solutions (where W denotes the weight filtration on de Rham cohomology). We also describe a procedure for finding solutions of these Picard-Fuchs equations.

Original languageEnglish (US)
Pages (from-to)8365-8389
Number of pages25
JournalTransactions of the American Mathematical Society
Issue number12
StatePublished - Dec 2020

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics


  • Hypergeometric series
  • Newton polytope
  • Picard-Fuchs equation


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