Let $p$ and $q$ be polynomials (maybe in several variables, over a field), and suppose they have $m$ and $n$ non-zero terms respectively. We can assume $m\leq n$. Can it ever happen that the product $p\cdot q$ has fewer than $m$ non-zero terms?
I ask this because I vaguely recall seeing a positive answer in book somewhere (probably about computation or algorithms since the polynomials were unwieldy). If anyone knows what book this is from it would be much appreciated.
Best Answer
$$(x^2-2x+2)(x^2+2x+2)=x^4+4.$$