An Introduction to Upper Half Plane Polynomials

For example, x1 + · · · + xd ∈ Ud (C). This follows from the fact that the upper half plane is a cone, so if σ1, . . . , σd are in the upper half plane then so is their sum. Another example is x1x2 − 1. If σ1 and σ2 are in the upper half plane then σ1σ2 ∈ C \ (0,∞), so σ1σ2 − 1 is not zero. U1 (C) is easily described. It is all polynomials in one variable whose roots are either real, or lie in the lower half plane. It is important to observe that the zero polynomial is not in U (C). This is unfortunate, since it causes many conclusions to be of the form “. . .∈ U (C) ∪ {0}. . . ”. Conventions: d is a positive integer, ı = √ −1, and y is a variable distinct from x1, . . . , xd. We use the following notation