Is the system uniquely solvable?

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Discussion Overview

The discussion revolves around the uniqueness of solutions for the system of equations $$x^3+y^3+z^3=1 \\ x\cdot y\cdot z=-1$$ in the vicinity of the point $(1, 1, 1)$. Participants explore whether this system can be uniquely solved for $y$ and $z$ as functions of $x$, and seek guidance on how to verify this property.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant questions whether the point $(1, 1, 1)$ is appropriate, noting that it does not lie on the surfaces defined by the equations, suggesting the point $(-1, 1, 1)$ instead.
  • Another participant corrects their previous statement, indicating they meant the point $(1, -1, 1)$.
  • A suggestion is made to apply the implicit function theorem if "region" is interpreted as "neighbourhood" to check for unique solvability.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct point for analysis, and the discussion remains unresolved regarding the uniqueness of the solution.

Contextual Notes

There is ambiguity regarding the appropriate point to analyze, as well as the implications of using the implicit function theorem, which may depend on the definitions and assumptions made about the region in question.

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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