Fixed-points of folded polynomial systems

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  1. Background:

  2. This page reviews algorithms and Mathematica code used to numerically compute fixed-points of folded polynomial systems.

    Consider a polynomial system $F(x,y)=\{p_1(x,y),p_2(x,y)\}$ where $p_1$ and $p_2$ are polynomials with rational coefficients. A point $(x,y)$ is a fixed-point of $F(x,y)$ if $F(x,y)=(x,y)$. For example, let $$ F(x,y)=\left\{\begin{array}{c} 1/2+3/4x+1/3y^2 \\ 2/5+1/2x^2-4/5xy \end{array}\right\} $$ Then a fixed-point of $F(x,y)$ is a point $(x_f,y_f)$ such that $F(x_f,y_f)={x_f,y_f}$. That is, one in which $1/2+3/4 x_f+1/3 y_f=x_f$ and $2/5+1/2 x_f^2-4/5 x_f y_f=y_f$. This is equivalent to finding the roots of $F(x,y)-\{x,y\}=\{0,0\}$

    And by Bezout's Theorem, we would expect to find four zeros to this system. It's easy to find four solutions in Mathematica via

    Mathematica code 2

    NSolve[{1/2 + 3/4 x + 1/3 y^2 - x == 0,2/5 + 1/2 x^2 - x y - y == 0}, {x, y}]
    which returns $$ \left( \begin{array}{cc} 2.29344\, -1.67169 i & 0.864008\, -0.725555 i \\ 2.29344\, +1.67169 i & 0.864008\, +0.725555 i \\ -0.793439+0.441414 i & -0.114008-1.45192 i \\ -0.793439-0.441414 i & -0.114008+1.45192 i \\ \end{array} \right) $$

    What happens if we fold $F(x,y)$? That is, the composition $F\circ F=F^{((2))}=F(F(x,y))$? We then have the system $$ F^{((2))}(x,y)=\left\{\frac{1}{3} \left(\frac{x^2}{2}-x y+\frac{2}{5}\right)^2+\frac{3}{4} \left(\frac{3 x}{4}+\frac{y^2}{3}+\frac{1}{2}\right)+\frac{1}{2},-\left(\frac{x^2}{2}-x y+\frac{2}{5}\right) \left(\frac{3 x}{4}+\frac{y^2}{3}+\frac{1}{2}\right)+\frac{1}{2} \left(\frac{3 x}{4}+\frac{y^2}{3}+\frac{1}{2}\right)^2+\frac{2}{5}\right\} $$ An easy way to visualize this is to let $u=1/2+3/4 x+1/3 y^2$ and $v=2/5+1/2 x^2-4 x y$ then $F^{((2))}(x,y)=F(u,v)$. And by Bezout's Theorem, the maximum number of roots to $F^{((2))}(x,y)-\{x,y\}$ would be $16$ which again can be easily computed in Mathematica. However, this method has computational limitations when $n$ becomes greater than about $10$ for the quadratic case in two variables and more limited when either the degree of the polynomials or the number of variables is increased.

    The central question of this section then is how do we study the fixed-points of $$ F^{((n))}(X)=\underbrace{F(F(F(\cdots F))\cdots)}_{\text{n times}} $$ with $X=\{x_1,x_2,\cdots,x_k\}$ when $n$ is large say $n=100$ and the polynomials are higher than degree $2$ with more than two variables?

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