This paper shows that the set of non-integers among rational numbers can be described by a single polynomial equation that uses only seven quantified rational unknowns. More precisely, for the rational numbers Q the author produces a polynomial P(t,x1,...,x7) with integer coefficients such that a rational number t is not an integer exactly when there exist rational x1,...,x7 solving P(t,x1,...,x7)=0. This improves the previous record of ten unknowns down to seven. The result is stated more generally for any global field (either a number field or a function field over a finite field).

What the paper proves in general form is Theorem 1.1: for any global field K and any finite set S0 of places (valuations) there is a polynomial F_{K,S0}(X,Y1,...,Y7) with coefficients in K such that an element x of K lies in the ring of S0-integers if and only if for all y1,...,y7 in K the polynomial F_{K,S0}(x,y1,...,y7) is nonzero. For K=Q and S0 empty this gives an integer-coefficient polynomial witnessing that Z (the integers) is definable in Q using seven universal quantifiers (equivalently the complement Q\Z is diophantine with seven existential quantifiers).

How do the authors do this? At a high level they use algebraic objects called quaternion algebras and two associated functions: the reduced norm and the reduced trace. These are the quaternion analogue of determinant and trace for 2×2 matrices. The key local idea is a short polynomial relation that produces two quaternion elements of reduced norm one whose reduced traces add up to a prescribed value c. At places where the quaternion algebra is ramified, integrality properties of these traces force c to lie in the local valuation ring. Putting these local constraints together in a global way gives a polynomial condition that detects S0-integers. To make the argument work uniformly and with few quantified variables the paper uses several technical steps: a parameterized form of Hensel’s lemma (a local lifting tool), a finite “freezing” of certain parameters at the ramified places, and a Hasse-principle type global argument to move from local solutions to a global one.