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  "body": "\n## Summary\n\nA monomial ideal is an ideal of a polynomial ring over a field that is generated by monomials.\nMembership in a monomial ideal is a combinatorial question about exponent vectors, so many invariants of these ideals can be computed by counting.\nEvery ideal can be deformed to a monomial ideal, its ideal of leading terms with respect to a [[monomial order|E874NARWPV]].\nThis is the bridge from general ideals to combinatorics through [[Gröbner bases|C83KPT47R0]].\n\n## Explanations\n\n### L1 human\n\nA monomial is a product of variables such as $x^2 y$.\nA monomial ideal is what one gets by taking some monomials and all their multiples, then all sums of those.\nPicture the exponents $(a, b)$ of the monomials $x^a y^b$ as points of a grid: a monomial ideal is a staircase region that is closed under moving up and to the right.\nFor example, $(x^2, xy, y^3)$ contains every monomial except $1, x, y, y^2$.\nA polynomial lies in the ideal exactly when each of its terms lies in the staircase.\n",
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