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Let $f(x)$ be degree $n$ polynomial, with $n+1$ nonzero monomial, i.e., all coefficients nonzero (for example if $n = 3$, then we could have $3x^3 + 2x^2 + x + 10$)

Let $g(x)$ be any polynomial of degree $m$, (which may have some coefficients equal to $0$; e.g., f.e, with $m = 4$, we could take $4x^4 + 2$

Let $h(x) = f(x) * g(x)$

What can be said about the number of terms in $h(x)$? Can it be less than the number of terms in f(x) or g(x)?

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Much is known, and much unknown, about this question. One reference is Andrzej Schinzel, On the reduced length of a polynomial with real coefficients, II, Funct. Approx. Comment. Math. Volume 37, Number 2 (2007), 445-459. The abstract goes, "The length $L(P)$ of a polynomial $P$ is the sum of the absolute values of the coefficients. For $P\in\mathbb{R}[x]$ the properties of $l(P)$ are studied, where $l(P)$ is the infimum of $L(PG)$ for $G$ running through monic polynomials over $\mathbb{R}$." projecteuclid.org/… –  Gerry Myerson May 5 '12 at 6:28
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1 Answer

Note that $$x^4+4=(x^2-2x+2)(x^2+2x+2).$$

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How about if $f(x) = (x - a)^n$ and $g(x)$ is still a random polynomial? Can $h(x) = f(x) * g(x)$ have smaller number of terms than $f(x)$? –  user30620 May 4 '12 at 19:59
    
Would have to think about it. The example $x^4+4$ was instantaneous because variants come up in contest math. For example, observe that we can now see that the only positive integer $x$ such that $x^4+4$ is prime is $x=1$. –  André Nicolas May 4 '12 at 20:45
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