Abstract
Let P(z) be a polynomial in one complex variable, with complex coefficients, and let z1,…,zn be its zeros. Assume, by normalization, that P(0)=1. The direct path from 0 to the root zj is the set {P(tzj),0≤t≤1}. We are interested in the altitude of this path, which is |P(tzj)| . We show that there is always a zero towards which the direct path declines near 0, which means |P(tzj)|<|P(0)| if t is small enough. However, starting with degree 5, there are polynomials for which no direct path constantly remains below the altitude 1.