The Descartes Rule of Sweeps and the Descartes Signature

The Descartes Rule of Signs, which establishes a bound on the number of positive roots of a polynomial with real coefficients, is extended to polynomials with complex coefficients. The extension is modified to bound the number of complex roots in a given direction on the complex plane, giving rise to the Descartes Signature of a polynomial. The search for the roots of a polynomial is sometimes aided by the following result, often taught in high school: Theorem 1 (The Descartes Rule of Signs). Let p(x) := a0x0 + a1x1 + a2x2 + · · ·+ anx be a polynomial with real coefficients a0, a1, . . . , an, all non-zero, and integer exponents 0 ≤ m0 < m1 < · · · < mn. The number of positive roots of p is then at most the number of sign changes in the coefficient sequence. More specifically, the number of positive roots differs from the number of sign changes by an even number. Discussion and proof of the Rule of Signs can be found in the mathematical literature dating back to Descartes’ own work in 1637, as well as online. As fascinating and elegant as the Rule may be, it never seemed entirely satisfying. One learns early on in mathematics that even the study of quadratic polynomials isn’t “complete” without consideration of non-real complex numbers, yet this iconic element of polynomial lore all but ignores them. As we show, making up for this oversight requires only re-thinking sign changes in a manner that ties in quite nicely with the conceptually-illuminating revelation that “multiplication by a negative” in arithmetic corresponds to a geometric half-turn about the origin of the complex plane. Unfortunately, our proof of the adapted Rule of Signs relies on Descartes’ original. Therefore, while the rotational view of complex multiplication makes the arithmetic intuitive, our new interpretation of the Rule of Signs does not (yet) provide any “ah-ha!” insights into the result. Date: Originally drafted 23 September, 2007. Updated 14 March, 2008, with changes in some notation, correction of some (but probably not all) errors, considerable simplification of the statement of the Rule of Sweeps itself, new images, and an epilog referencing the related submission in the Wolfram Demonstration Project. Updated 21 April, 2008, with additional changes in notation, a once-again-revised statement of the Rule, and explicit sweep formulas. 1In our notation, the traditional (lowest-power) “trailing term” comes first and (highest-power) “leading term” comes last; we will refer to them as the “initial term” and “final term”, respectively. 2See, for instance, http://www.cut-the-knot.org/fta/ROS2.shtml 3Of course, using the Rule, one can occasionally glean some information about the number of non-real roots: subtract the maximum number of positive and (after a standard trick) negative roots from the polynomial’s degree. 1 2 B.D.S. “DON” MCCONNELL 1. The Rule of Sweeps 1.1. Preliminaries. Let (1) r(x) := c0x0 + c1x1 + · · ·+ cnx be a polynomial with complex coefficients, all non-zero, and integer exponents 0 ≤ m0 < m1 < · · · < mn. We can write r(x) = p(x) + i · q(x), where (2) p(x) := a0x0 + a1x1 + · · ·+ anx q(x) := b0x0 + b1x1 + · · ·+ bnx and ck = ak + ibk with real ak and bk for each k. Clearly, any positive root of r is a positive root of both p and q, so our analysis of r amounts to a tandem analysis of p and q via the Rule of Signs, with one provision: although the coefficient sequence {ck} is not all-zero, (at most) one of the sequences {ak} or {bk} might be, defying attempts to count sign changes; we shall therefore agree that Agreement. The number of sign changes in an all-zero coefficient sequence is infinite. As any number whatsoever is a root of an identically-zero polynomial, this Agreement allows us to preserve the conclusion that the number of roots is no greater than the number of coefficient sign changes, and we may write (3) # of positive roots of r ≤ min (# of sign changes in {ak},# of sign changes in {bk}) Our strategy is to compute an upper bound on right-hand side of (3), using the coefficient sequence {ck}. To do this, we introduce sweeps. 1.2. Sweeps defined. Imagine a needle with one end anchored at the origin of the complex plane and with the other end initially pointing in the direction of c0. Let the free end of the needle sweep —always counter-clockwise— to point in the directions of c1, c2, etc., and finally cn, “stalling in place” when successive coefficients are identical, and tracing out a “sweep spiral”. (See Figure 1.) We define the positive sweep of the polynomial as the total angle swept by the counter-clockwise needle, computing it thusly: (4) sweep(r) := n ∑ k=1 arg(ck/ck−1) where 0 ≤ arg(z) < 2π Likewise, we define the negative sweep of the polynomial as (the absolute value of) the total angle swept by the needle if it were to move always clockwise. This is equivalent to a counter-clockwise sweep with the needle taking the coefficients in reverse order: (5) sweep−(r) := n ∑