Supertropical quadratic forms I

Abstract

We initiate the theory of a quadratic form q over a semiring R. As customary, one can write q(x+y)=q(x)+q(y)+b(x,y), where b is a companion bilinear form. But in contrast to the ring-theoretic case, the companion bilinear form need not be uniquely defined. Nevertheless, q can always be written as a sum of quadratic forms q=κ+ρ, where κ is quasilinear in the sense that κ(x+y)=κ(x)+κ(y), and ρ is rigid in the sense that it has a unique companion. In case that R is a supersemifield (cf. Definition 4.1 below) and q is defined on a free R-module, we obtain an explicit classification of these decompositions q=κ+ρ and of all companions b of q. As an application to tropical geometry, given a quadratic form q:V→R on a free module V over a commutative ring R and a supervaluation ρ: R→U with values in a supertropical semiring [5], we define - after choosing a base L=(vi|i∈I) of V- a quadratic form qφ:U(I)→U on the free module U(I) over the semiring U. The analysis of quadratic forms over a supertropical semiring enables one to measure the “position” of q with respect to L via φ.