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  "body": "\n## Statement\n\nLet $p$ be a nonzero polynomial with real coefficients, and let $l \u003c r$ be real numbers.\nFor $c \\in \\R$, let $V_c$ be the number of sign changes in the sequence of nonzero coefficients of the polynomial $p(x + c)$.\nLet $N$ be the number of [[roots|00EHX6FWC0]] of $p$ in the half-open interval $(l, r]$, counted with [[multiplicity|03XVP8Q4GY]].\nThen $N \\le V_l - V_r$, and $V_l - V_r - N$ is even.\n\n## Remarks\n\nThe roots of $p(x + c)$ are the roots of $p$ shifted by $-c$.\nSo [[Descartes' rule of signs|T6DQQ36JZ5]] applied to $p(x + l)$ and $p(x + r)$ bounds the roots of $p$ greater than $l$ and greater than $r$.\nThe theorem says that the difference of the two bounds is again a bound, with the same parity rule.\nExample: take $p = x^2 - 2$, $l = 0$ and $r = 2$.\nThen $p(x) = x^2 - 2$ has $V_0 = 1$, and $p(x + 2) = x^2 + 4x + 2$ has $V_2 = 0$.\nSo there is exactly one root in $(0, 2]$, namely $\\sqrt 2$.\nTake $l = 0$ and $r$ larger than the real part of every complex root of $p$.\nThen all coefficients of $p(x + r)$ have the same sign, so $V_r = 0$, and one gets back Descartes' rule.\n\n## Explanations\n\n### L1 human\n\nBudan's theorem bounds the number of roots of a polynomial in an interval by counting sign changes of coefficients.\nOne shifts the polynomial so that each end of the interval moves to $0$, and then compares the two counts.\nLike Descartes' rule, it can overcount, but only by an even number.\n\n### L2 human\n\nThe theorem is used in root isolation: an interval with $V_l - V_r = 0$ has no root, and one with $V_l - V_r = 1$ has exactly one.\nAn equivalent form, due to Fourier, counts sign changes in the sequence $p(c), p'(c), \\dots, p^{(n)}(c)$.\nBy Taylor's formula, these numbers are the coefficients of $p(x + c)$ up to positive factors.\nCompared with [[Sturm's theorem|R44CV0KD9T]], it is cheaper to compute but gives only a bound.\n",
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