# Degree inequality of a polynomial map distinguishing hyperplanes

Let $$H_1, \ldots, H_n$$ be $$n$$ linearly independent hyperplanes in $$k^n$$, for some arbitrary field $$k$$. Let $$X = H_1 \cup H_2 \cup \cdots \cup H_n$$. Is it true that if $$F=(f_1, \ldots, f_n)$$ is a polynomial map from $$k^n$$ to $$k^n$$, such that $$F(X) \cap F(k^n - X) = \emptyset$$, then $$\sum \deg(f_i) \ge n$$?

This holds under the stronger condition that for all $$a \in X$$, $$F(a)$$ has at least one coordinate equal to zero, and for all $$a \notin X$$, $$F(a)$$ has all coordinates nonzero: $$\prod f_i$$ then cuts out $$X$$, but since any polynomial cutting out $$X$$ has degree at least $$n$$, the conclusion follows.

More generally, for a variety $$X \subseteq k^n$$, define $$C(X)$$ to be the minimum of the sum of the degrees of the coordinate functions over all polynomial maps $$F$$ where $$F(X) \cap F(k^n - X) = \emptyset$$. Is this quantity equivalent to something that's well known? You trivially have $$C(X) \le n$$ by taking $$F$$ to be the identity map. Also, if $$X$$ is defined by equations whose degree-sum equals $$m$$, $$C(X) \le m$$. Is one of these inequalities always sharp?

Proceed by induction on $$n$$. By your reasoning it holds for $$n = 1$$. For arbitrary $$n \geq 2$$, take a generic hyperplane $$H$$ distinct from the $$H_1, \ldots, H_m$$, and apply induction on the restriction of $$F$$ to $$H$$.
• I'm having trouble understanding the induction. Say $n=m=2$. Restricting $F$ to a generic line through the origin, by the $n=1$ case the sum of the degrees of the restriction is at least 1. But you want to say this sum is at least 2. How do you get the +1? Jun 18 '20 at 0:40
• $m$ does not change. Since $H$ is generic, $H'_j := H \cap H_j$ will be nonempty for each $j$. Linear independence of $H_1, \ldots, H_m$ in $k^n$ "should" imply linear independence of $H'_1, \ldots, H'_m$ in $k^{n-1}$. In particular if $n = m = 2$, after restricting to a generic line you will have $n = 1$ and $m = 2$. Jun 18 '20 at 15:47
• How can you have 2 linearly independent points in $k$? Jun 18 '20 at 16:00