The Positive Dominance Algorithm

In this note I will explain a computational algorithm which I call the positive dominance algorithm. The input is a polynomial F ∈ R[X1, ..., Xn] and a polytope P ⊂ R. One version of the algorithm tries to verify that F > 0 on P . This version halts if and only if the the assertion is true. Another version of the algorithm tries to verify that F ≥ 0 on P . This case is more interesting, because it allows for sharper applications. In this case, we have F ≥ 0 on P if the algorithm halts, but the converse is not necessarily true. The algorithm will probably fail if F = 0 on some interior points of the polytope. The algorithms require that P has some triangulation into simplices, so we will restrict our attention to the case when P is a simplex. In case F is a rational polynomial and P is a rational simplex, the algorithm can be implemented with exact rational arithmetic. Another useful case happens when both P and F are defined over the same number field. Here, again, the algorithm can be implemented using exact arithmetic. I have used the algorithm in two diverse situations. In one situation [S1], I proved some inequalities converning the Cayley-Menger determinant on the space of tetrahedra, and in another situation [S2] I used it to analyze the Julia set of a high degree rational map of 2 variables. I’m sure it has many other applications. After presenting the algorithm, I’ll discuss the geometric applications I have in mind.