Few Product Gates but Many Zeroes

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 offer the remote possibility of being helpful for factoring integers and because their existence for infinitely many d would disprove a form of the Blum-CuckerShub-Smale conjecture (strengthened to allow arbitrary constants). A natural step towards a better understanding of the BCSS conjecture would thus be to construct d-gems or to rule out their existence. Ruling out d-gems for large d is currently totally out of reach. Here the best we can do towards that goal is to prove that skew 2n-gems if they exist require n {+,−}-gates and that skew 2n-gems for any n ≥ 5 would provide new solutions to the Prouhet-Tarry-Escott problem in number theory (skew meaning the further restriction that each {+,−}-gate merely adds an integer to a polynomial). In the opposite direction, here we do manage to construct skew d-gems for several values of d up to 55.