MHB Is the system uniquely solvable?

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The discussion centers on the solvability of the system of equations $$x^3+y^3+z^3=1$$ and $$x\cdot y\cdot z=-1$$ near the point $(1, -1, 1)$. Participants clarify that the original point $(1, 1, 1)$ is not on the surfaces defined by the equations. A suggestion is made to apply the implicit function theorem to determine if the system is uniquely solvable for $y$ and $z$ as functions of $x$. The focus remains on the mathematical approach to check the conditions for unique solvability. The conversation emphasizes the importance of correctly identifying the point of interest in relation to the equations.
mathmari
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Hey! :o

Is the system $$x^3+y^3+z^3=1 \\ x\cdot y\cdot z=-1$$ in a region of the point $(1; 1; 1)$ uniquely solvable for $y = y (x) $ and $z = z (x)$ ?

How can we check that? Could you give me a hint? (Wondering)
 
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mathmari said:
Hey! :o

Is the system $$x^3+y^3+z^3=1 \\ x\cdot y\cdot z=-1$$ in a region of the point $(1; 1; 1)$ uniquely solvable for $y = y (x) $ and $z = z (x)$ ?

How can we check that? Could you give me a hint? (Wondering)
The point $(1,1,1)$ does not lie on either of those surfaces. Did you mean the point $(-1,1,1)$?
 
Opalg said:
The point $(1,1,1)$ does not lie on either of those surfaces. Did you mean the point $(-1,1,1)$?

Oh I meant $(1;-1;1)$. (Tmi)
 
mathmari said:
Hey! :o

Is the system $$x^3+y^3+z^3=1 \\ x\cdot y\cdot z=-1$$ in a region of the point $(1; 1; 1)$ uniquely solvable for $y = y (x) $ and $z = z (x)$ ?

How can we check that? Could you give me a hint? (Wondering)

If by "region" you mean "neighbourhood", then my hint would be to use the implicit function theorem.
 

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