Few product gates but many zeros

Abstract

A d-gem is a {+,-,×}-circuit having very few ×-gates and computing from {x}∪ℤ a univariate polynomial of degree d having d distinct integer roots. We introduce d-gems because they could help factoring integers and because their existence for infinitely many d would blatantly disprove a variant of the Blum-Cucker-Shub-Smale conjecture. A natural step towards validating the conjecture would thus be to rule out d-gems for large d. Here we construct d-gems for several values of d up to 55. Our 2 n -gems for n≥4 are skew, that is, each {+,-}-gate adds an integer. We prove that skew 2 n -gems if they exist require n {+,-}-gates, and that these for n≤5 would imply new solutions to the Prouhet-Tarry-Escott problem in number theory. By contrast, skew d-gems over the real numbers are shown to exist for every d. © 2009 Springer Berlin Heidelberg.