If $P$ is a polynomial over the field $\mathbb F_q$ of degree at most $q-2$ with $k$ nonzero coefficients, then $P$ has at most $(1-1/k)(q-1)$ distinct nonzero roots.
Does this fact have any standard name / reference / proof / refinements / extensions?
Note that the number of distinct non-zero roots of a polynomial $P(x)$ over $\mathbb{F}_q$ always equals to the minimal degree of a non-zero polynomial belonging to the ideal generated by $P$ and $x^{q-1}-1$. This is why the following argument looks natural and I expect something similar to appear in other extensions of this result.
Denote $P(x)=\sum_{i=1}^k c_i x^{a_i}$, $0=a_1<a_2<a_3<\dots<a_k\leqslant q-2<a_{k+1}:=q-1$. Choose $i\in \{1,2,\dots,k\}$ such that $a_{i+1}-a_i\geqslant (q-1)/k$ and reduce the polynomial $Q(x)=x^{q-1-a_{i+1}}P(x)$ modulo $x^{q-1}-1$. The remainder $R(x)$ has the same non-zero roots as $P$, but $\deg R(x)\leqslant \max (q-1+a_i-a_{i+1},a_k-a_{i+1})=q-1+a_i-a_{i+1}\leqslant (q-1)(1-\frac1k)$.
This method allows to get something for the polynomials in many variables. Say, if a non-zero polynomial $P(x_1,\dots,x_n)$ has degree at most $q-2$ in each variable and has at most $M$ non-zero-coefficients, we may estimate the number of points $a=(a_1,\dots,a_n)\in (\mathbb{F}_q^*)^n$ for which $P(a)=0$. Namely, choose some ``forbidden'' set $\Omega\in \{0,1,\dots,q-2\}^n$ and look for a monomial $x_1^{c_1}\dots x_n^{c_n}$ for which all monomials of the reduced polynomial $x_1^{c_1}\dots x_n^{c_n}P(x_1,\dots,x_n)$ (reduced modulo the ideal $\langle x_1^{q-1}-1,\dots,x_n^{q-1}-1\rangle$, that does not change the zeroes in $(\mathbb{F}_q^*)^n$) do not belong to $\Omega$. If we choose random exponents $c_1,\dots,c_n$, each specific monomial belongs to $\Omega$ after multiplying by $x_1^{c_1}\dots x_n^{c_n}$ and reduction with probability $|\Omega|/(q-1)^n$. Thus if $M\cdot |\Omega|<(q-1)^n$, we may avoid $\Omega$ by suitable multiplication and reduction. For certain choices of $\Omega$ this gives some upper bounds on the number of zeroes by De Millo -- Lipton -- Schwartz -- Zippel -- Alon -- Füredi -- $\dots$ theory.
This is just an uncertainty principle for the cyclic group $\mathbb F_q^*$; see, e.g. this paper by Tao.
