On polynomial values of anisotropic quadratic forms

Abstract

Let k be a field, chark≠2, let φ be an anisotropic quadratic form over k, dimφ≥2, V the underlying linear space of φ. As usual, denote by D(φ) the set of nonzero values of φ. Given a positive integer m, we say that φ is m-essential if there exists a nonzero polynomial p∈k[x1,…,xm] such that p∈D(φk(x1,…,xm)), but p∉D(ψk(x1,…,xm))) for any anisotropic form ψ over k with dimψ<dimφ. We say that φ is strongly m-essential if additionally p=φ(u) for some vector u∈V⊗kk[t1,…,tm]. Among other results we show that if chark=0, b∈k⁎, and the form π≃《−1,b》 is anisotropic, then π is 2-essential. If, additionally, −3∈k⁎, then π is strongly 2-essential. We prove also that any anisotropic 2-fold Pfister form over k becomes strongly 2-essential over a certain field extension of k. To prove these statements we consider the family of polynomialspa,b,ω(x,y)=−bx4y2+b2x2y4−b(a+2ω)x2y2−(ω+a)416a in two variables over k, which are generalizations of the Motzkin polynomial f(x,y)=x2y4+x4y2−3x2y2+1, since f(x,y)=p−1,−1,−1(x,y). We show that pa,b,ω∈D(《a,b》k(x,y)), but, on the other hand, p−1,b,−1∉D(φ)k(x,y) for each 3-dimensional anisotropic form φ over k, provided chark=0. The result is a generalization of a theorem of Cassels, Ellison and Pfister, which claims that the Motzkin polynomial f(x,y) is not a sum of three squares over R(x,y).Finally, at the end of the paper we give examples showing that the Cassels-Pfister theorem in two variables does not hold in general for n-dimensional anisotropic forms provided n≥4.