## Abstract

We initiate the theory of a quadratic form q over a semiring R, with a view to study tropical linear algebra. As customary, one can write q(x + y) = q(x) + q(y) + b(x,y), where b is a companion bilinear form. In contrast to the classical theory of quadratic forms over a field, the companion bilinear form need not be uniquely defined. Nevertheless, q can always be written as a sum of quadratic ...

## Abstract

We initiate the theory of a quadratic form q over a semiring R, with a view to study tropical linear algebra. As customary, one can write q(x + y) = q(x) + q(y) + b(x,y), where b is a companion bilinear form. In contrast to the classical theory of quadratic forms over a field, the companion bilinear form need not be uniquely defined. Nevertheless, q can always be written as a sum of quadratic forms q = q(QL) + rho, where q(QL) is quasilinear in the sense that q(QL)(x y) = q(QL)(x) + q(QL)(y), and rho is rigid in the sense that it has a unique companion. In case that R is supertropical, we obtain an explicit classification of these decompositions q = q(QL) + rho and of all companions b of q, and see how this relates to the tropicalization procedure. (C) 2015 Elsevier B.V. All rights reserved.