Supertropical Quadratic Forms I

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 $q = \kappa + \rho,$ where $\kappa$ is quasilinear in the sense that $\kappa(x+y) = \kappa(x) + \kappa(y),$ and $\rho$ 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 $q = \kappa + \rho$ and of all companions $b$ of $q$. As an application to tropical geometry, given a quadratic form $q: V \to R$ on a free module $V$ over a commutative ring $R$ and a supervaluation $\varphi:R \to U$ with values in a supertropical semiring [5], we define - after choosing a base $L=(v_i | i\in I)$ of $V$ - a quadratic form $q^\varphi: U^{(I)} \to U$ 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 $\varphi$.