Let $f_0, f_1, \dots, f_k$ be $n$--variable polynomials
over a finite prime field $\fp$. A proof of the ideal
membership $f_0 \in \langle f_1, \dots, f_k \rangle$ in {\em
polynomial calculus} is a sequence of polynomials $h_1, \dots,
h_t$ such that $h_t = f_0$, and such that every $h_i$ is
either an $f_j$, $j \geq 1$, or obtained from $h_1, \dots,
h_{i-1}$ by one of the two inference rules: $g_1$ and $g_2$
entail any $\fp$--linear combination of $g_1$, $g_2$, and $g$
entails $g \cdot g'$, for any polynomial $g'$. The degree of
the proof is the maximum degree of the $h_i$'s. We give a
condition on families $\{ f_{N,0}, \dots,
f_{N,k_N}\}_{N<\omega}$ of $n_N$--variable polynomials of
bounded degree implying that the minimum degree of polynomial
calculus proofs of $f_{N,0}$ from $f_{N,1}, \dots, f_{N,k_N}$
cannot be bounded by an independent constant and, in fact, is
$\Omega(\log(\log(N)))$. In particular, we obtain an
$\Omega(\log(\log(N)))$ lower bound for the degrees of proofs
of $1$ (so called {\em refutations}) of the $(N,m)$--system
(defined in \cite{BIKPP}) formalizing a modular counting
principle (where $m$ is fixed and not divisible by $p$, and
the parameter $N$ is not divisible by $m$), and a similar
lower bound for refutations of systems encoding that $N$ is
composite (whenever $N$ is a prime). No bounds were previously
known for these systems. The same method yields
$\Omega(\log(N))$ lower bounds for the degree of coefficient
polynomials in Nullstellensatz proofs. The method is based on
a new result about a uniform way of generating all submoduli
of tabloid moduli.