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  "body": "\n## Summary\n\nA regular chain is a triangular set of polynomials, with one polynomial for each of some of the variables.\nIn it, the leading coefficient of each polynomial is not a zero divisor modulo the ideal described by the earlier polynomials.\nRegular chains describe the solution sets of polynomial systems in a triangular form that can be solved one variable at a time.\nEvery polynomial system can be decomposed into finitely many regular chains, and this gives an alternative to [[Gröbner bases|C83KPT47R0]] in computer algebra.\n\n## Explanations\n\n### L1 human\n\nA triangular system first constrains $x$, then $y$ in terms of $x$, then $z$ in terms of $x$ and $y$, and so on.\nTo solve it, one solves for $x$, substitutes, solves for $y$, and continues.\nThis fails if the coefficient of the highest power of $y$ vanishes for some allowed values of $x$.\nA regular chain is a triangular system in which this never happens in a harmful way.\nEach leading coefficient is invertible, or at least not a zero divisor, given the earlier equations.\nFor example, $x^2 - 1 = 0$, $xy - 1 = 0$ is a regular chain, because $x$ is nonzero whenever $x^2 = 1$.\n",
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