  An integer polynomial $p$ of $n$ variables is called a \emph{threshold gate}
for a Boolean function $f$ of $n$ variables if for all $x \in \zoon$ $f(x)=1$
if and only if $p(x)\geq 0$. The \emph{weight} of a threshold gate is the sum
of its absolute values.
  In this paper we study how large a weight might be needed if we fix some
function and some threshold degree. We prove $2^{\Omega(2^{2n/5})}$ lower bound
on this value. The best previous bound was $2^{\Omega(2^{n/8})}$ (Podolskii,
2009).
  In addition we present substantially simpler proof of the weaker
$2^{\Omega(2^{n/4})}$ lower bound. This proof is conceptually similar to other
proofs of the bounds on weights of nonlinear threshold gates, but avoids a lot
of technical details arising in other proofs. We hope that this proof will help
to show the ideas behind the construction used to prove these lower bounds.
