A coincidence theorem for symmetric multilinear forms

Abstract

We obtain a coincidence theorem for (vector-valued) symmetric multilinear forms which generalizes an important classical result due to J. L. Walsh, commonly known as “Walsh's coincidence theorem” on symmetric n-linear forms in complex variables. Our main theorem, being a storehouse of many applications (as is Walsh's theorem in the classical theory), provides new results as well as a large number of known results due to Walsh, Zervos, Marden, Hörmander, Szegö, and Zaheer (some in improved forms). We also give a number of examples in support of certain claims that we make about our main theorem.