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Let $V$ be an algebraic variety in $C^n$ and when $k\ge 0$ is an integer then $\mathrm{Pol}(V,k)$ denotes all holomorphic functions $f\left(z\right)$ on $V$ satisfying $\left|f\left(z\right)\right|\le C_f(1+|z|{)}^k$ for all $z\in V$ and some constant $C_f$. We estimate the least integer $ϵ(V,k)$ such that every $f\in \mathrm{Pol}(V,k)$ admits an extension from $V$ into $C^n$ by a polynomial $P(z_1,...,z_n)$, of degree $k+ϵ(V,k)$ at most. In particular ${lim}_{k\>\infty }ϵ(V,k)$ is related to cohomology groups with coefficients in coherent analytic sheaves on $V$. The existence of the finite integer $ϵ(V,k)$ is for example an easy consequence of Kodaira’s Vanishing Theorem.