Say you have an affine variety $X$ in $n$-dimensional affine space. (You can even assume we are over $\mathbb{C}$, but I believe the nature of my question is algebraic).

I want to bound from above the complexity of a system of equations defining $X$. My guess is that the following parameters should be enough:

  1. $n$ - dimension of ambient space.

  2. $\mu$ - dimension of $X$

  3. $d$ - degree of $X$ in say the standart projective compactification of $\mathbb{A}^n$.

I don't mind what concept of complexity to take, say the max power in which any variable is taken from any equation.

Thank you very much!


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