What are some applications of Gröbner bases that could be interesting to a group of students that more or less only studied Chapter 1 and Chapter 2 of Ideals, Varieties, and Algorithms by David A. Cox John, Little and Donal O'Shea? They are ex-National Olympiad medalists or even IMO and IOI contenders, so they have very good mathematical intuition but they do not know much about college maths other than the notions in those 2 chapters + very little abstract algebra and calculus (which they did in the first year of college).
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1Does this answer your question? Applications of Gröbner bases – David G. Stork May 27 '23 at 21:44
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1@DavidG.Stork No. Most of those, except the one related to sudoku involve much complicated notions for proofs. – johnyy May 27 '23 at 21:44
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1What about the application of determining if two ideals are equal using reduced Gröbner bases? – wormram May 27 '23 at 21:58
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See for example Using Gröbner bases for solving polynomial equations, and an example of solving an equation with radicals. – dxiv May 27 '23 at 22:51
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@morrowmh Hmm, could you give me a reference to that ? – johnyy May 28 '23 at 17:07
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@johnyy Chapter 2, Section 7 of Ideals, Varieties, and Algorithms by Cox, Little, O'Shea. There they talk about reduced bases and their uniqueness. – wormram May 28 '23 at 19:46
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1How about the application of finding a generating set for the kernel of a ring homomorphism between polynomial rings? Especially if you cast it as the problem of finding polynomial relations between polynomials. e.g. for $x = t^2$, $y = tu$, $z = u^2$, you find the relation $xz = y^2$ not by inspection but by a mechanical calculation. – Daniel Schepler Jun 01 '23 at 21:38