Given a system of complex polynomial equations in $n$ variables, giving rise to an affine variety $V \subseteq \mathbb C^n$, is there a bound $b \in \mathbb R$ such that if $V(\mathbb C) \neq \emptyset$ then there is a point $a \in V(\mathbb C)$ such that $||a|| < b$?

I am looking for some bound $b$ in terms of $n$, the degrees of the polynomials and the absolute values of the coefficients. Or indeed is anything known in this sort of direction?

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