ag.algebraic geometry – complexity of the closure of the image of a morphism of algebraic sets

Let $F$ be a field. Define the complexity of an algebraic set $X$ over $F$ in $mathbb{A}^m$ to be the smallest integer $n>m$ s.t $X$ is the zero set of at most $n$ polynomials with degree at most $n$. If $phi$ is a morphism between 2 algebraic varayeties then it’s complexity is defined to be the complexity of it’s graph.

Suppose $X,Y$ are affine varayeties over a field $F$ and $phi:Xrightarrow Y$ is a morphism between them. Suppose that $X,Y,phi$ all have complexity at most $M$ for some positive integer $M$. can we bound the complexity of $overline{Imphi}$ with a bound depending only on $M$? (where the closure is of course zeriski closure).