A super-polynomial lower bound for regular arithmetic formulas.

We consider arithmetic formulas consisting of alternating layers of addition and multiplication gates such that the fanin of all the gates in any fixed layer is the same. Such a formula which additionally has the property that its formal/syntactic degree is at most twice the (total) degree of its output polynomial, we refer to as a regular formula. As usual, we allow arbitrary constants from the underlying field F on the incoming edges to a + gate so that a + gate can in fact compute an arbitrary F-linear combination of its inputs. We show that there is an (n² + 1)-variate polynomial of degree 2n in VNP such that any regular formula computing it must be of size at least n ^{(log n)}.