Solving a system of two polynomial equations

Question

Does the following system of two polynomial equations

$$\begin{align} x^2-y^2+4x-1=0\\ \tfrac{1}{2}x^3+xy^2+4y+1=0 \end{align}$$

have a solution in the unit disc $\{(x,y)\in\mathbb{R}^2\mid x^2+y^2\leq1\}$?

Intuitively, I think this problem is closely related to the inverse function theorem. But I don't know how to work this out. Can somebody help me? Thanks.

Answer 1

from your first equation we get $$y=\pm\sqrt{x^2+4x-1}$$ plugging this in you second equation we obtain $$4(\pm\sqrt{x^2+4x-1})=-\frac{1}{2}x^3-1-x(x^2+4x-1)$$ squaring both sides we get $$-9/4\,{x}^{6}-12\,{x}^{5}-13\,{x}^{4}+5\,{x}^{3}+7\,{x}^{2}+66\,x-17=0$$ by a numerical method we get $$x\approx .2506835437, 1.295854035$$

Answer 2

You can prove the existance of a solution by using the intermediate value theorem, which is applicable because of the continuity of the functions involved. I will sketch the solution and you can fill in the details.

First, take the equation

$$x^2-y^2+4x-1=0$$

With some basic analysis, you can figure out that this equation describes a hyperbola, only one branch of which is intersecting the unit disk. Then, determine where this hyperbola intersects the boundary of the unit disk, i.e. solve

$$\begin{cases} x^2-y^2+4x-1=0\\ x^2+y^2=1 \end{cases}$$

which should not be too hard. Then evaluate the function $F(x,y)=\frac{1}{2}x^3+xy^2+4y+1$ in the two solutions from the previous equation. If all goes well, you should find a positive and a negative value. If one of the values is zero, you of course found the exact solution of the system.

But if you have a positive and negative value, you can argue by the intermediate value theorem that there is at least one point $(x^*,y^*)$ on the hyperbola, between the intersections with the boundary of the disk, such that $F(x^*,y^*)=0$.

And then, if you want to have a numerical value, you can do what Dr. Sonnhard Graubner did.

Here's a graphical visualisation:

Answer 3

Indeed, the given algebraic curves do intersect inside the unit disk.

Visual inspection of the plot above tells us that the intersection is close to $\left(\frac 14, -\frac 14\right)$, which can serve as an initial estimate. Using SymPy, we solve the given system of polynomial equations numerically:

>>> from sympy import *
>>> x = Symbol('x')
>>> y = Symbol('y')
>>> p1 = x**2 - y**2 + 4*x - 1
>>> p2 = x**3 + 2*x*y**2 + 8*y + 2
>>> print(nsolve((p1, p2), (x, y), (0.25, -0.25)))
[ 0.250683543718194]
[-0.256078921358018]