Real-rooted polynomials and a generalised Hermite-Sylvester theorem

A polynomial is real-rooted if all of its roots are real. For every polynomial $f(t)\in \mathbf{R}[t]$, the Hermite-Sylvester theorem associates a quadratic form ${\mathrm{\Phi }}_2$ such that $f(t)$ is real-rooted if and only if ${\mathrm{\Phi }}_2$ is positive semidefinite. In this note, for every positive integer $m$, an $2m$-adic form ${\mathrm{\Phi }}_{2m}$ is constructed such that $f(t)$ is real-rooted if and only if ${\mathrm{\Phi }}_{2m}$ is positive semidefinite for some $m$ if and only if ${\mathrm{\Phi }}_{2m}(x_1,\dots ,x_n)$ is positive semidefinite for all $m$.