# Switching lemma for polynomials over $\mathbb{F}_2$

Suppose $$f$$ is in $$\mathbb{F}_2[x_1,...,x_n]$$ with total degree $$d$$.

Q. Is there any kind of switching lemma or restriction lemma in which by applying the lemma on $$f$$ we can reduce the total degree of $$f$$ in a good way?

• Sorry if I'm missing something, but is there a reason this intuitively seems like it might be possible? For instance, when $f(x)=\sum_{S\subseteq [n]: \vert S\vert=d} x^S$, any restriction that lowers the degree would have to restrict at least $n-d+1$ variables. – J.G Jul 1 at 19:03
• @JasonGaitonde: I see. By your example, it seems there is no result of this kind for general polynomial, but it may be possible to have such a result if we have sparser polynomial than your example. You take all of the monomials of the $d$, but if we have some kind of reasonable bound on the number of the whole monomials, it may lead to a reasonable thing. – Erfan Khaniki Jul 2 at 20:14