Mathematics and Statistics Vol. 14(2), pp. 235 - 247
DOI: 10.13189/ms.2026.140205
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An Application of Holomorphic Dynamics to Solve Some Systems of Quadratic Equations

Gamaliel Blé ¹, Carlos Marín ¹, Rogelio Valdez ^2,*
1 Academic Division of Basic Sciences, Juárez Autonomous University of Tabasco, Cunduacán, 86690, Tabasco, México
² Research Center in Sciences, Autonomous University of the State of Morelos, Cuernavaca, 62209, Morelos, México

ABSTRACT

In the 2011 Mexican Mathematical Olympiad, contestants were asked to solve a cyclic system involving a real quadratic polynomial. In this paper, we extend that problem by using the dynamics of complex quadratic polynomials as an alternative method to determine the number of solutions of some cyclic systems of real quadratic equations with variables. The real quadratic polynomials associated with the systems are conjugated to polynomials of the form , with a real parameter. Using the behavior of the real orbits for quadratic polynomials of the form , standard complex dynamics tools, and the theory of orbit portraits, we determine how many real solutions these new cyclic systems of equations with polynomials have, depending on the position of the parameters in the real slice of the Mandelbrot set. We prove that if is a positive parameter outside the Mandelbrot set, then the system has no solutions; if we restrict our parameters to the main cardioid of the Mandelbrot set, then the system has at least one solution; in fact, there are only two solutions. When we move further to the left in the real line, we go through the doubling-period cascade of real hyperbolic components of the Mandelbrot set, where solutions always exist and the number of solutions of the system depends on the period of the real components, and the congruence of the number of equations in the system modulo that period. Finally, if , a system with equations always has a solution; moreover, for each positive divisor of , there will be an orbit of period for , which is a solution of the system. The main advantage of the technique used here, compared to, for example, Gaussian elimination algorithms or Gröbner bases theory, is that not only can we determine if the system has a solution or not, but we can also determine the different windows in the real line where the number of solutions stays stable, and where the number of solutions changes.

KEYWORDS
Equation Systems, Periodic Orbits, Quadratic Family, Mandelbrot Set

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Gamaliel Blé , Carlos Marín , Rogelio Valdez , "An Application of Holomorphic Dynamics to Solve Some Systems of Quadratic Equations," Mathematics and Statistics, Vol. 14, No. 2, pp. 235 - 247, 2026. DOI: 10.13189/ms.2026.140205.

(b). APA Format:
Gamaliel Blé , Carlos Marín , Rogelio Valdez (2026). An Application of Holomorphic Dynamics to Solve Some Systems of Quadratic Equations. Mathematics and Statistics, 14(2), 235 - 247. DOI: 10.13189/ms.2026.140205.