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TR11-094 | 20th June 2011 03:32
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#### Computing polynomials with few multiplications

**Abstract:**
A folklore result in arithmetic complexity shows that the number of multiplications required to compute some $n$-variate polynomial of degree $d$ is $\sqrt{{n+d \choose n}}$. We complement this by an almost matching upper bound, showing that any $n$-variate polynomial of degree $d$ over any field can be computed with only $\sqrt{{n+d \choose n}} \cdot (nd)^{O(1)}$ multiplications.