Enumerative geometry via the A^1-degree

Morel's $A^1$ -degree in $A^1$-homotopy theory is the analog of the Brouwer degree in classical topology. It takes values in the Grothendieck-Witt ring $GW(k)$ of a field $k$, that is the group completion of isometry classes of non-degenerate symmetric bilinear forms. We can use the $A^1$ -degree to count algebro-geometric objects in $GW(k)$, giving an $A^1$-enumerative geometry over non-algebraically closed fields. Taking the rank and the signature recovers classical counts over the complex and the real numbers, respectively. For example, the count of lines on a smooth cubic surface enriched in $GW(k)$ has rank 27 and signature 3.