Consider an algebraic subset $Vsubseteq mathbb{R}^{n+1}$ defined as the zero set of polynomials ${f_i}$ and the projection map $pi: mathbb{R}^{n+1}to mathbb{R}^n$ deleting the last entry.

By the Tarski-Seidenberg theorem, the image $pi(V)$ is a semi-algebraic set. Is there a bound on the degree of the polynomials characterizing $pi(V)$ in terms of the degrees of the polynomials $f_i$?