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Does the following system of six simultaneous equations in eight variables $x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4$ have solutions in $\mathbb{R}$? in $\mathbb{C}$? $$x_1y_2-x_2y_1=1$$ $$x_1y_3-x_3y_1=0$$ $$x_1y_4-x_4y_1=0$$ $$x_2y_3-x_3y_2=0$$ $$x_2y_4-x_4y_2=0$$ $$x_3y_4-x_4y_3=1$$

Maple cannot help; I don't know if Mathematica will.

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The system is underdetermined on purpose? –  J. M. Nov 28 '11 at 17:44
    
That is correct. It is underdetermined. –  TCL Nov 28 '11 at 17:48
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I'm killing the nonlinear tag because, well, the vast majority of things in life are nonlinear. –  Willie Wong Nov 30 '11 at 10:23
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1 Answer

up vote 2 down vote accepted

There is no solution. (Check my work.)

First note that $x_1\ne 0$ as if it was, either $y_1=0$ or $x_3$ and $x_4=0$, and both of those would cause violations. Now note that $x_3\ne 0$, as that would force either $x_1$ or $y_3$ to be 0, again vioplating the constraints. Similarly, (but more convolutedly $x_4\ne 0$. So we get $$y_1 = \frac{x_1 y_3}{x_3}$$ $$y_2 = \frac{x_2 y_3}{x_3}$$ as $x_3\ne 0$.

Plugging $y_1$ and $y_2$ into the first equation we get $0=1$, so the answer is no in either field.

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