Agrawal, Kayal, and Saxena recently introduced a new method of proving that an integer is prime. The speed of the Agrawal-Kayal-Saxena method depends on proven lower bounds for the size of the multiplicative semigroup generated by several polynomials modulo another polynomial h. Voloch pointed out an application of the Stothers-Mason ABC theorem in this context: under mild assumptions, distinct...

We consider a new notion of circuit uniformity based on the concept of rational relations, called Rational-uniformity and denoted Rat. Our goal is to prove it is sharper than DLogTime-uniformity, the notion introduced by Barrington et al. [3], denoted DLT , that is: 1) Rational-uniformity implies DLogTime-uniformity, 2) we have NCRat NCDLT ( NCRat, 3) we have ∀k > 0, NCDLT ⊆ NC Rat , 4) Rationa...

We use the method of interlacing families polynomials to derive a simple proof Bourgain and Tzafriri’s Restricted Invertibility Principle, then sharpen result in two ways. show that stable rank can be replaced by Schatten 4-norm tighter bounds hold when number columns matrix under consideration does not greatly exceed its rows. Our are derived from an analysis smallest zeros Jacobi associated L...