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I'm doing an assignment about Groebner basis and it's applications and I want to show how it helps us to solve a system of polynomials equations, I've been working with the book "An Introduction to Computational Algebraic Geometry and Commutative Algebra" of David Cox, John Little, Donal O’She, the problem is that the book doesn't cover why Groebner basis helps us to solve a system of polynomials equation, they talk briefly why Buchberger’s algorithm is a general algorithm for row-reduction algorithm, and after that they give some examples to how to use Groebner basis to solve a system of polynomials equation but without explain why does it works, I've tried to read it in wikipedia but also there it's not written why does it works.

I'll be glad if someone could give me sketch of proof or idea why does it work, i.e. why if we works with the lexicographic order on $\mathbb{Z}_{\geq 0}^{n}$ and compute a basis for ideal that generated by $n$ polynomials (and the system has finitely many solutions) then we get a polynomial only depends in $x_n$ which lies in the basis ?

Thank you all in advance!

user26857
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oneneedsanswers
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    The basic idea is that if $J$ is an ideal of the ring of polynomials from $k[x_1,x_2,\ldots,x_n]$ then the set of common zeros of all the polynomials in $J$ is the same as the set of common zeros of any collection of polynomials that generates the ideal $J$. Typically $J$ is defined by some generating set that we want to replace with a Gröbner basis (=a generating set of a special form). For the purposes of finding the zeros, a Gröbner basis works much the same way like any linear system with a triangular matrix of coefficients - emphasis in elimination of the variables in some order. – Jyrki Lahtonen Aug 22 '24 at 18:01
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    @Moo I see the corollary I need but the proof is in "An Introduction to Grobner Bases" of Adams, but I couldn't find the proof in the first section of the book (which I assumed the proof of that corollary should be there. – oneneedsanswers Aug 22 '24 at 18:39
  • @JyrkiLahtonen Actually I understood the first part (I saw that proposition in the course I took, but I can't understand why Groebner basis works the same way like any linear system with a triangular matrix of coefficients.. – oneneedsanswers Aug 22 '24 at 18:41
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    Related. I can post a few examples, but it sounds like you have seen enough already. Don't have a proof at hand, my copy of von zur Gathen & Gerhard is in my office, and I'm not sure whether they have it? – Jyrki Lahtonen Aug 23 '24 at 18:41
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    @JyrkiLahtonen Yeah, I've seen that post. After hours of thinking and searching I've found that there is a theorem (Theorem 2 in Section 3 Chapter 1 in the book I mentioned in the post) which does a reduction to prove that if ideal $I$ has dimension zero then $I \cap \mathbb{C}[x_n]$ isn't the empty set and that's a commutative algebra theory theorem (can be found in "A Singular Introduction to Commutative Algebra" theorem 3.5.1), all of that proves that if a system of polynomial equations has finitely many solutions then we can do that procedure. But thank you for your help though. – oneneedsanswers Aug 23 '24 at 19:52
  • I'm not familiar with the specific reference you're using. But at a high level, the analogy between Groebner bases and row echelon form would be a theorem such as: "Let $A$ be a matrix in row echelon form. Then for any nonzero element of the row space of $A$, its first nonzero entry is in the same column as one of the pivots of $A$. i.e. There is some row of $A$ such that its first nonzero entry is in the same column as the first nonzero entry of the given vector in the row space of $A$." ... – Daniel Schepler Aug 23 '24 at 21:54
  • This "first nonzero entry column position" would correspond to initial monomial of a polynomial in the Groebner basis side of the analogy. And then, to extend the analogy, if you want to determine whether a vector is in the row space or not, you can use that theorem to repeatedly subtract a multiple of a row to keep shifting the first nonzero entry to the right. Until eventually, you either reach zero and the original vector was in the row space; or else, you reach a point where the first nonzero entry does not match any row, and... – Daniel Schepler Aug 23 '24 at 21:56
  • you can conclude the original vector was not in the row space. – Daniel Schepler Aug 23 '24 at 21:56

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