  Let $f,g_1,\dots,g_m$ be polynomials of degree at most $d$ with real
coefficients in a vector of variables $x=(x_1,\dots,x_n)$. Assume that $f$ is
non-negative on a basic semi-algebraic set $S$ defined by polynomial
inequalities $g_j(x)\ge 0$, for $j=1,\dots,m$. Our previous work
[arXiv:2205.04254 (2022)] has stated several representations of $f$ based on
the Fritz John conditions. This paper provides some explicit degree bounds
depending on $n$, $m$, and $d$ for these representations. In application to
polynomial optimization, we obtain explicit rates of finite convergence of the
hierarchies of semidefinite relaxations based on these representations.
