Few Product Gates But Many Zeros

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-gems for n ≤ 4 are skew, that is, each {+,−}-gate adds an integer. We prove that skew 2-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.