‎we investigate graham higman's paper enumerating $p$-groups‎, ‎ii‎, ‎in which he formulated his famous porc conjecture‎. ‎we are able to simplify some of the theory‎. ‎in particular‎, ‎higman's paper contains five pages of homological algebra which he uses in‎ ‎his proof that the number of solutions in a finite field to a finite set of‎ ‎monomial equations is porc‎. ‎it turns out tha...

We prove a strong form of the Brumer–Stark Conjecture and, as a consequence, a strong form of Rubin’s integral refinement of the abelian Stark Conjecture, for a large class of abelian extensions of an arbitrary characteristic p global field k. This class includes all the abelian extensions K/k contained in the compositum kp∞ := kp · k∞ of the maximal pro-p abelian extension kp/k and the maximal...

We give a polynomial counterexample to a discrete version of the Markus-Yamabe Conjecture and a conjecture of Deng, Meisters and Zampieri, asserting that if F : C → C is a polynomial map with det(JF ) ∈ C∗, then for all λ ∈ R large enough λF is global analytic linearizable. These counterexamples hold in any dimension ≥ 4.

In this paper, we prove that most of the boolean functions, f : {−1, 1}n → {−1, 1} satisfy the Fourier Entropy Influence (FEI) Conjecture due to Friedgut and Kalai (Proc. AMS’96)[2]. The conjecture says that the Entropy of a boolean function is at most a constant times the Influence of the function. The conjecture has been proven for families of functions of smaller sizes. O’donnell, Wright and...

We give a polynomial counterexample to both the Markus-Yamabe Conjecture and the discrete Markus-Yamabe problem for all dimensions ≥ 3.