Which Exterior Powers are Balanced?

Which Exterior Powers are Balanced?

A signed graph is a graph whose edges are given ±1 weights. In such a graph, the sign of a cycle is the product of the signs of its edges. A signed graph is called balanced if its adjacency matrix is similar to the adjacency matrix of an unsigned graph via conjugation by a diagonal ±1 matrix. For a signed graph Σ on n vertices, its exterior kth power, where k = 1, . . . , n−1, is a graph ∧k Σ w...

Exterior Powers

Let R be a commutative ring. Unless indicated otherwise, all modules are R-modules and all tensor products are taken over R, so we abbreviate ⊗R to ⊗. A bilinear function out of M1 × M2 turns into a linear function out of the tensor product M1 ⊗ M2. In a similar way, a multilinear function out of M1 × · · · ×Mk turns into a linear function out of the k-fold tensor product M1 ⊗ · · · ⊗Mk. We wil...

On sums which are powers

Which Powers of Holomorphic Functions Are Integrable?

Question 1. Let f(z1, . . . , zn) be a holomorphic function on an open set U ⊂ C. For which t ∈ R is |f |t locally integrable? The positive values of t pose no problems, for these |f |t is even continuous. If f is nowhere zero on U then again |f |t is continuous for any t ∈ R. Thus the question is only interesting near the zeros of f and for negative values of t. More generally, if h is an inve...

On shifted products which are powers

Fermat gave the first example of a set of four positive integers {a1, a2, a3, a4} with the property that aiaj + 1 is a square for 1 ≤ i < j ≤ 4. His example was {1, 3, 8, 120}. Baker and Davenport [1] proved that the example could not be extended to a set of 5 positive integers such that the product of any two of them plus one is a square. Kangasabapathy and Ponnudurai [6], Sansone [9] and Grin...