# 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).