We study real bihomogeneous polynomials r ( z , stretchy="false">¯<!-- ¯ ...

We consider cones of real forms which are sums squares and invariant under a (finite) reflection group. Using the representation theory these groups we able to use symmetry inherent in give more efficient descriptions. focus especially on An, Bn, Dn case where so-called higher Specht polynomials uniform description cones. These descriptions allow us, deduce that fixed degree 2d stabilizes with ...

We construct families of explicit polynomials f over Q that are sums of squares of polynomials over R, but not over Q. Whether or not such examples exist was an open question originally raised by Sturmfels. We also study representations of f as sums of squares of rational functions over Q. In the case of ternary quartics, we prove that our counterexamples to Sturmfels’ question are the only ones.