The Polynomial Model in the Study of Counterexamples to S. Piccard’s Theorem

The search for homometric structures, i.e., non-congruent structures sharing the same autocorrelation function, is shown to be of a combinatorial nature and can be studied using purely algebraic techniques. Several results on the existence of certain homometric structures which contradict a theorem by S. Piccard are proved based on a polynomial representation model and the factorization of polynomials over the rationals. Combinatorial arguments show that certain factorizations do not lead to counterexamples to S. Piccard’s theorem.