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  "body": "\n## Summary\n\nFix a [[monomial order|E874NARWPV]] on a polynomial ring.\nA Gröbner basis of an ideal is a finite generating set whose leading terms generate the ideal of all leading terms of the ideal.\nWith a Gröbner basis, the [[multivariate division algorithm|T8EN3CAEQF]] has a unique remainder.\nSo one can decide whether a polynomial lies in the ideal, and one can compute in the quotient ring.\nEvery ideal has a Gröbner basis, and [[Buchberger's algorithm|4RH1D40WG1]] computes one.\nGröbner bases are the basic tool of computational commutative algebra and algebraic geometry, including [[elimination theory|9PA8CVSWPD]].\n\n## Explanations\n\n### L1 human\n\nFor polynomials in one variable, an ideal is generated by one polynomial, and division by it decides membership.\nIn several variables, a set of generators may be badly chosen: a polynomial in the ideal can have a leading term that none of the generators' leading terms divides.\nA Gröbner basis is a set of generators without this defect.\nFor example, $\\set{x + y, x}$ generates the ideal $(x, y)$.\nBut $y = x - (x + y)$ is in the ideal, and for the order with $x \u003e y$ its leading term $y$ is not divisible by $x$.\nSo this set is not a Gröbner basis, while the set $\\set{x, y}$ is.\n",
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