Pointers for direct proof of extension of the Descartes Rule of Signs to complex polynomials?

2010-04-25T14:52:02Z

The following describes an extension of the Descartes Rule of Signs to polynomials with complex coefficients.

First, I need to define the notion of a "sweep"... Given a complex polynomial p(z) := c₀ z^m₀ + c₁ z^m₁ + ... + c_n z^m_n with non-zero c_i and strictly increasing m_i, let {α_i} be the coefficients' "non-decreasing argument sequence": α_i = arg c_i (mod 2π), and α₀ ≤ α₁ ≤ ... ≤ α_n, with each α_i+1-α_i taken as small as possible; then α_n-α₀ computes the "angular sweep" --which I'll denote sweep(p)-- of a needle with one end anchored at the origin of the Complex Plane that begins pointing at c₀ and then spins counter-clockwise to point at each c_i in turn ("stalling in place" when consecutive coefficients have equal arguments).[*]

With this, we have:

The Descartes Rule of Sweeps. The number of positive real roots of p is at most $\lfloor \frac{1}{\pi} sweep(p)\rfloor$.

(The Descartes Rule of Signs represents a special case: each sign change in a polynomial's real coefficient sequence contributes π to the sweep, so that $\frac{1}{\pi}sweep(p)$ exactly counts those sign changes.)

Now, I can prove the Rule of Sweeps using the Descartes Rule of Signs itself, but that approach sheds no light on why the Rule of Signs works (which is what I'm really after). The result seems to be one clever contour away from falling directly out of the Cauchy Argument Principle --and it even seems curiously appropriate that the formula provides a kind of half-winding number from the coefficient sequence-- but apparently I'm not sufficiently clever. :(

Suggestions?

[*] Note: One can also compute a sweep of the coefficients taken in reverse order (or, equivalently, spinning the needle clockwise). Differently-directed sweeps are usually not equal, so we can optimize the bound in the Rule of Sweeps by taking "the" sweep as the minimum of the two directed sweeps.

Answer by Victor Miller for Pointers for direct proof of extension of the Descartes Rule of Signs to complex polynomials?

2010-04-26T03:04:18Z

The following paper by Jeff Lagarias and Tom Richardson "Multivariate Descartes rule of signs and Sturmfels' challenge problem" http://www.math.lsa.umich.edu/~lagarias/doc/intelligencer.pdf

looks like it has things relevant to your question.

Answer by Greg for Pointers for direct proof of extension of the Descartes Rule of Signs to complex polynomials?

2010-09-30T18:24:06Z

This paper is related to generalizing Sturm's theorem rather than Descartes' rule of signs, but I think it is relevant: http://arxiv.org/abs/0808.0097

Pointers for direct proof of extension of the Descartes Rule of Signs to complex polynomials? - MathOverflow

Pointers for direct proof of extension of the Descartes Rule of Signs to complex polynomials?

Answer by Victor Miller for Pointers for direct proof of extension of the Descartes Rule of Signs to complex polynomials?

Answer by Greg for Pointers for direct proof of extension of the Descartes Rule of Signs to complex polynomials?