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Consider the system of polynomial equations \begin{align} x + y + z &= a,\\ xy &= b,\\ z&=c, \end{align} where $x, y, z$ are the unknowns, and $a,b,c$ are known real numbers. It is clear at a first glance that there exist at least two solutions to the system: the variables $x$ and $y$ can be exchanged.

I don't understand how I can use Gröbner bases to decide (in a more general case) on whether a system of polynomial equations may have multiple solutions. The Gröbner basis for the system above is $\{z-c,y^2+(c-a)y+b,x+y+c-a\}$ (computed with Mathematica). How would I use the Gröbner basis to conclude?

(Example taken from this article)

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  • $\begingroup$ You are right about the second part, the first not really because it does not have a unique solution, does it? $\endgroup$
    – G. Gare
    Commented Dec 9, 2022 at 13:46
  • $\begingroup$ @DietrichBurde I have modified the question according to your comment. Thanks! $\endgroup$
    – G. Gare
    Commented Dec 9, 2022 at 13:48
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    $\begingroup$ Note that Groebner bases depend on a choice of monomial order (so one wouldn't say THE Groebner basis without specifying an order). $\endgroup$ Commented Dec 9, 2022 at 13:56

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A Groebner basis is given by $$ \mathcal{G}=\{ xy - b,a - x - y - z,c - z\}, $$ and the solutions in $\Bbb C(a,b,c)$ are given by $$ (x,y,z)=(x,a-c-x,c), $$ where $x$ is one of the two roots of $$ x^2+(a+c)x+b=0. $$ This follows directly from the equations.

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