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  "body": "\n## Summary\n\nThe multivariate division algorithm divides a polynomial in several variables by an ordered list of polynomials, with respect to a fixed [[monomial order|E874NARWPV]].\nIt writes the polynomial as a combination of the divisors plus a remainder.\nNo term of the remainder is divisible by the leading term of a divisor.\nIt extends the [[division algorithm for polynomials|0MES3KS0ZC]] in one variable.\nIn general the remainder depends on the order of the divisors, but for a [[Gröbner basis|C83KPT47R0]] it is unique, and it then decides ideal membership.\n\n## Explanations\n\n### L1 human\n\nIn one variable, long division removes the leading term of the dividend again and again until the degree is too small.\nWith several variables one does the same.\nWhenever the leading term of what is left is a multiple of the leading term of some divisor, subtract a suitable multiple of that divisor.\nWhen no divisor fits, move the leading term to the remainder and go on.\nFor example, dividing $xy^2 + 1$ by $xy + 1$ and then $y + 1$ gives $xy^2 + 1 = y(xy + 1) - (y + 1) + 2$, with remainder $2$.\n",
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