Frontiers in Mathematical Sciences

A Dichotomy Principle for Representations of Primes by Quadratic Forms

Mohammad Gholamzadeh Mahmoudi
Sharif University of Technology


The quadratic value theorem provides a dichotomy principle for an anisotropic quadratic form $q$ over a given field $F$ and an irreducible polynomial $p$ in a polynomial ring $A$ in finitely many variables over $F$:
(1) either $q$ remains anisotropic over the residue field of $A$ at $p$,
(2) or a scalar multiple of $p$ lies in the group generated by the nonzero values represented by $q$ over $A$.
In this talk, we examine the validity of this principle in a wider context.