The number of positive real roots of a polynomial with real coefficients is strictly smaller than the number of nonzero coefficients of the polynomial. This is an immediate corollary of Descartes' rule of signs.

For a polynomial with complex coefficients of degree at most $p-1$, with $p$ prime, the number of distinct roots of the polynomial which are degree-$p$ roots of unity is strictly smaller than the number of nonzero coefficients of the polynomial. As observed by Tao, this is equivalent to an uncertainty inequality for prime-order groups.

For a polynomial over a field of zero characteristic, the multiplicity of any of its non-zero roots is strictly smaller than the number of nonzero coefficients of the polynomial. To my knowledge, this first appeared in a paper by Brindza.

Are these results reducible to each other? (Well, the ground fields are not quite the same, and yet...) Are there any other similar results known? And, ultimately, is there any "common parent" from which all these results can be derived?

sharpbound of $q^{m-1}$ on the number of distinct roots of a polynomial with $m$ non-zero coefficients: see the paperZeros of sparse polynomials over local fields of characteristic $p$, by Bjorn Poonen. Lenstra's reference there addresses the $\mathbb{Q}_p$ variant, but with a non-sharp bound. $\endgroup$