Solution to a non-linear simultaneous equation

I'm trying to solve the non-linear, two-variable system of equations \begin{align*} y-x+x^5-\frac{y^4x}{2(1+x^2)^2}-\frac{x^3}{1+y^2} &= 0 \\ -x-y+y^5-\frac{x^4y}{2(1+y^2)^2}-\frac{y^3}{1+x^2} &= 0 \end{align*} with $$x,y \in \mathbb{R}$$. You might notice that the second equation is almost obtained from the first by swapping $$x$$ and $$y$$, except for the first term whose signs are opposed. I think the only solution is the origin $$x=y=0$$, and this is seems to be confirmed numerically on WolframAlpha, but I'm not sure, and I don't know how to prove it. You can also write the system as \begin{align*} y-x+x^5+\frac{1}{4}\partial_x\left(\frac{y^4}{1+x^2}-\frac{x^4}{1+y^2}\right) &= 0 \\ -x-y+y^5-\frac{1}{4}\partial_y\left(\frac{y^4}{1+x^2}-\frac{x^4}{1+y^2}\right) &= 0 \end{align*} Any hints/ideas?

• The only solution is x=y=0 (desmos.com/calculator/fkxeyiwyks). Apr 17 '20 at 15:48
• As I said, this definitely seems true from numerical experiments, but how can I prove it? Apr 17 '20 at 15:50

I converted both equations into polynomials and put them into Singular to find all solutions.

LIB "solve.lib";
ring r=0,(x,y),dp;
poly p1=2x9y2+2x9+4x7y2+2x7-4x5+2x4y3+2x4y-4x3y2-6x3+4x2y3+4x2y-xy6-xy4-2xy2-2x+2y3+2y;
poly p2=-x6y-x4y-2x3y4-4x3y2-2x3+2x2y9+4x2y7-4x2y3-2x2y-2xy4-4xy2-2x+2y9+2y7-4y5-6y3-2y;
ideal i=p1,p2;
def T=solve(i,30);


There are $$89$$ distinct solutions, but $$88$$ of them involve a complex number.

This shows that the last solution, $$x=y=0$$, is the only real solution.

• Thanks for this. Is this library mathematically proven to yield all solutions? Apr 17 '20 at 16:00
• @chaos Yes. Because it works with Gröbner bases and other exact polynomial root-finding schemes, it is one of the most powerful mathematical libraries around. You should give it a try. Apr 17 '20 at 16:06
• Thank you, I was not even aware that exact polynomial root-finding algorithms existed. Is it possible to give exact expressions for all roots, or is it only possible to determine the number of real roots, which is sufficient here? Apr 17 '20 at 16:09
• @chaos Abel-Ruffini theorem: you cannot determine exact expressions for quintic+ polynomials in the general case. Simply counting roots is sufficient. Apr 17 '20 at 16:10
• Yes, that's what I thought. Thanks again! Apr 17 '20 at 16:13

According to Maple, the resultant of the numerators of the left sides of the two equations with respect to $$y$$ is $$2\,{x}^{9} \left( 4096\,{x}^{80}+69632\,{x}^{78}+516096\,{x}^{76}+ 2084864\,{x}^{74}+4272128\,{x}^{72}-163840\,{x}^{70}-25825280\,{x}^{68 }-65273856\,{x}^{66}-38729728\,{x}^{64}+154712064\,{x}^{62}+382149632 \,{x}^{60}+144474112\,{x}^{58}-791342976\,{x}^{56}-1359673728\,{x}^{54 }+226609664\,{x}^{52}+3532913024\,{x}^{50}+4419130368\,{x}^{48}- 672814592\,{x}^{46}-8622736384\,{x}^{44}-11440427968\,{x}^{42}- 5943181984\,{x}^{40}+2080361568\,{x}^{38}+6219084784\,{x}^{36}+ 8104484032\,{x}^{34}+13320834677\,{x}^{32}+21495544525\,{x}^{30}+ 25938523882\,{x}^{28}+22524511317\,{x}^{26}+14404728925\,{x}^{24}+ 7384009128\,{x}^{22}+3941633218\,{x}^{20}+2832052128\,{x}^{18}+ 2199851588\,{x}^{16}+1418048480\,{x}^{14}+701995632\,{x}^{12}+ 263885888\,{x}^{10}+74884576\,{x}^{8}+15715584\,{x}^{6}+2310144\,{x}^{ 4}+208896\,{x}^{2}+8192 \right) \left( {x}^{2}+1 \right) ^{6}$$ The only real root of this is $$x=0$$, which implies $$y=0$$ from the first equation.

• Thanks for this. How do you prove that this polynomial of order $80$ has no real roots? Apr 17 '20 at 16:01
• With Maple's sturm command, which uses Sturm's theorem. Apr 17 '20 at 16:06
• Thank you, I was not aware of this theorem. Unfortunately I can only accept one of the two answers, but I will upvote both... I don't know what the policy is here, but I will accept the other simply for providing a little more explanation in the comment, though your answer is equally flawless... Apr 17 '20 at 16:13
The text passage contains a mathematical question regarding the solution to a system of non-linear equations, and it provides an answer that the only real solution is \(x = y = 0\).
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Solution to a non-linear simultaneous equation

I'm trying to solve the non-linear, two-variable system of equations \begin{align*} y-x+x^5-\frac{y^4x}{2(1+x^2)^2}-\frac{x^3}{1+y^2} &= 0 \\ -x-y+y^5-\frac{x^4y}{2(1+y^2)^2}-\frac{y^3}{1+x^2} &= 0 \end{align*} with $$x,y \in \mathbb{R}$$. You might notice that the second equation is almost obtained from the first by swapping $$x$$ and $$y$$, except for the first term whose signs are opposed. I think the only solution is the origin $$x=y=0$$, and this is seems to be confirmed numerically on WolframAlpha, but I'm not sure, and I don't know how to prove it. You can also write the system as \begin{align*} y-x+x^5+\frac{1}{4}\partial_x\left(\frac{y^4}{1+x^2}-\frac{x^4}{1+y^2}\right) &= 0 \\ -x-y+y^5-\frac{1}{4}\partial_y\left(\frac{y^4}{1+x^2}-\frac{x^4}{1+y^2}\right) &= 0 \end{align*} Any hints/ideas?

• The only solution is x=y=0 (desmos.com/calculator/fkxeyiwyks). Apr 17 '20 at 15:48
• As I said, this definitely seems true from numerical experiments, but how can I prove it? Apr 17 '20 at 15:50

I converted both equations into polynomials and put them into Singular to find all solutions.

LIB "solve.lib";
ring r=0,(x,y),dp;
poly p1=2x9y2+2x9+4x7y2+2x7-4x5+2x4y3+2x4y-4x3y2-6x3+4x2y3+4x2y-xy6-xy4-2xy2-2x+2y3+2y;
poly p2=-x6y-x4y-2x3y4-4x3y2-2x3+2x2y9+4x2y7-4x2y3-2x2y-2xy4-4xy2-2x+2y9+2y7-4y5-6y3-2y;
ideal i=p1,p2;
def T=solve(i,30);


There are $$89$$ distinct solutions, but $$88$$ of them involve a complex number.

This shows that the last solution, $$x=y=0$$, is the only real solution.

• Thanks for this. Is this library mathematically proven to yield all solutions? Apr 17 '20 at 16:00
• @chaos Yes. Because it works with Gröbner bases and other exact polynomial root-finding schemes, it is one of the most powerful mathematical libraries around. You should give it a try. Apr 17 '20 at 16:06
• Thank you, I was not even aware that exact polynomial root-finding algorithms existed. Is it possible to give exact expressions for all roots, or is it only possible to determine the number of real roots, which is sufficient here? Apr 17 '20 at 16:09
• @chaos Abel-Ruffini theorem: you cannot determine exact expressions for quintic+ polynomials in the general case. Simply counting roots is sufficient. Apr 17 '20 at 16:10
• Yes, that's what I thought. Thanks again! Apr 17 '20 at 16:13

According to Maple, the resultant of the numerators of the left sides of the two equations with respect to $$y$$ is $$2\,{x}^{9} \left( 4096\,{x}^{80}+69632\,{x}^{78}+516096\,{x}^{76}+ 2084864\,{x}^{74}+4272128\,{x}^{72}-163840\,{x}^{70}-25825280\,{x}^{68 }-65273856\,{x}^{66}-38729728\,{x}^{64}+154712064\,{x}^{62}+382149632 \,{x}^{60}+144474112\,{x}^{58}-791342976\,{x}^{56}-1359673728\,{x}^{54 }+226609664\,{x}^{52}+3532913024\,{x}^{50}+4419130368\,{x}^{48}- 672814592\,{x}^{46}-8622736384\,{x}^{44}-11440427968\,{x}^{42}- 5943181984\,{x}^{40}+2080361568\,{x}^{38}+6219084784\,{x}^{36}+ 8104484032\,{x}^{34}+13320834677\,{x}^{32}+21495544525\,{x}^{30}+ 25938523882\,{x}^{28}+22524511317\,{x}^{26}+14404728925\,{x}^{24}+ 7384009128\,{x}^{22}+3941633218\,{x}^{20}+2832052128\,{x}^{18}+ 2199851588\,{x}^{16}+1418048480\,{x}^{14}+701995632\,{x}^{12}+ 263885888\,{x}^{10}+74884576\,{x}^{8}+15715584\,{x}^{6}+2310144\,{x}^{ 4}+208896\,{x}^{2}+8192 \right) \left( {x}^{2}+1 \right) ^{6}$$ The only real root of this is $$x=0$$, which implies $$y=0$$ from the first equation.

• Thanks for this. How do you prove that this polynomial of order $80$ has no real roots? Apr 17 '20 at 16:01
• With Maple's sturm command, which uses Sturm's theorem. Apr 17 '20 at 16:06
• Thank you, I was not aware of this theorem. Unfortunately I can only accept one of the two answers, but I will upvote both... I don't know what the policy is here, but I will accept the other simply for providing a little more explanation in the comment, though your answer is equally flawless... Apr 17 '20 at 16:13
user
The text passage contains a mathematical question regarding the solution to a system of non-linear equations, and it provides an answer that the only real solution is \(x = y = 0\).

TYPE: SINGLE_QUESTION_WITH_ANSWER