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  "body": "\n## Summary\n\nA monomial order is a total order on the monomials of a polynomial ring that is compatible with multiplication and has no infinite descending chains.\nIt singles out a leading term in every nonzero polynomial, which is what makes the [[multivariate division algorithm|T8EN3CAEQF]] and [[Gröbner bases|C83KPT47R0]] work.\nCommon examples are the lexicographic order, the graded lexicographic order and the graded reverse lexicographic order.\n\n## Explanations\n\n### L1 human\n\nFor polynomials in one variable, terms are ordered by degree, and the leading term is the one of highest degree.\nWith several variables, there is no single natural choice, so one fixes a rule that compares any two monomials, such as $x^2 y$ and $x y^3$.\nThe rule must respect multiplication: if $u$ comes before $v$, then $wu$ comes before $wv$.\nIt must also make $1$ the smallest monomial, so that repeatedly cancelling leading terms always stops.\nIn the lexicographic order with $x \u003e y$, one compares the powers of $x$ first, so $x^2 y \u003e x y^3$.\n",
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