An analysis of the solution set to a homotopy equation between polynomials with real coefficients

Abstract

We investigate the structure of the solution setS to a homotopy equationH(Z,t)=0 between two polynomialsF andG with real coefficients in one complex variableZ. The mapH is represented asH(x+iy, t)=h 1(x, y, t)+ih 2(x, y, t), whereh 1 andh 2 are polynomials from ℝ2 × [0,1] into ℝ and i is the imaginary unit. Since all the coefficients ofF andG are real, there is a polynomialh 3 such thath 2(x, y, t)=yh3(x, y, t). Then the solution setS is divided into two sets {(x, t)∶h 1(x, 0, t)=0} and {(x+iy, t)∶h 1(x, y, t)=0,h 3(x, y, t)=0}. Using this division, we make the structure ofS clear. Finally we briefly explain the structure of the solution set to a homotopy equation between polynomial systems with real coefficients in several variables.