## David Eisenbud

The *Ulrich Complexity” of a homogeneous form F(x_0,\dots, x_n) is the smallest integer d such that F^d can be written as the determinant of a matrix of linear forms. Thus, for example the Ulrich Complexity of the determinant polynomial itself is 1, as is the Ulrich complexity of any irreducible polynomial in 3 variables over an algebraically closed field; but for a quadratic form of rank n the Ulrich complexity is at least 2^{n/2-2}. The Ulrich complexity of every homogeneous form is known to be finite—already a nontrivial result. I’ll describe some of what’s known and connect this invariant to other problems, from Valiant’s conjecture to elimination theory and resolutions over complete intersections.