[Paper Review] New Multiplier Sequences via Discriminant Amoebae

Mikael Passare, Boris Shapiro+1 more|arXiv (Cornell University)|Jan 24, 2010

Topological and Geometric Data Analysis6 references3 citations

TL;DR

This paper introduces two new classes of polynomials—sign-independently real-rooted and their nonnegative coefficient variants—using discriminant amoebae and amoeba theory. It characterizes multiplier sequences preserving these classes via recession cones of amoebae, showing that such sequences satisfy explicit linear inequalities derived from the geometry of discriminants.

ABSTRACT

In their classic 1914 paper, Polya and Schur introduced and characterized two types of linear operators acting diagonally on the monomial basis of R[x], sending real-rooted polynomials (resp. polynomials with all nonzero roots of the same sign) to real-rooted polynomials. Motivated by fundamental properties of amoebae and discriminants discovered by Gelfand, Kapranov, and Zelevinsky, we introduce two new natural classes of polynomials and describe diagonal operators preserving these new classes. A pleasant circumstance in our description is that these classes have a simple explicit description, one of them coinciding with the class of log-concave sequences.

Motivation & Objective

• To extend classical multiplier sequence theory by introducing new classes of polynomials preserved under diagonal linear operators.
• To characterize multiplier sequences preserving sign-independently real-rooted polynomials using the geometry of amoebae and discriminants.
• To provide an explicit, geometric characterization of multiplier sequences of the third kind through recession cones of amoebae of discriminants.
• To offer a new, geometric framework for understanding multiplier sequences using tools from algebraic geometry, particularly amoebae and the Horn-Kapranov uniformization.
• To address open problems in the theory of real-rooted polynomials by linking them to discriminant amoebae and their connected components.

Proposed method

• Uses the amoeba of the discriminant of the polynomial family $\Delta_k$ to define a geometric domain $\Gamma_k'$, which captures the image of the logarithmic absolute value map on sign-independently real-rooted polynomials.
• Applies the Horn-Kapranov uniformization to parametrize the $A$-discriminant variety, enabling explicit computation of the recession cone $C_k'$ of the amoeba complement.
• Derives a system of linear inequalities $kx_j \leq j(x_k - x_0)$ for $j = 1, \dots, k-1$ that characterize the recession cone $C_k'$, which corresponds to the multiplier sequences of the third kind.
• Establishes a diffeomorphism $\mathrm{Log}|\cdot|: II^\geq_k \to \Gamma_k'$, showing that the image of the nonnegative sign-independently real-rooted polynomials under the log-absolute value map is precisely the connected component $\Gamma_k'$.
• Employs Descartes’ Rule of Signs and discriminant formulas for trinomials to verify boundary behavior of test polynomials $p_j^\pm_{\varepsilon}$, ensuring the necessity of the derived inequalities.
• Proves sufficiency by showing that any sequence $\gamma$ satisfying the inequalities lies in the recession cone $C_k'$, hence preserving $II^\geq_k$ under the diagonal operator $T_\gamma$.

Experimental results

Research questions

• RQ1 What geometric structure underlies the set of multiplier sequences preserving sign-independently real-rooted polynomials?
• RQ2 Can the class of multiplier sequences of the third kind be characterized via the amoeba of the discriminant of the polynomial family $\Delta_k$?
• RQ3 How do the connected components of the complement of the reflected discriminant relate to the number of multiplier sequence classes?
• RQ4 Is there a purely algebraic or combinatorial proof of the characterization of multiplier sequences of the third kind, avoiding advanced amoeba theory?
• RQ5 What is the precise relationship between the recession cone of the amoeba of $\Delta_k$ and the necessary and sufficient conditions for a sequence to preserve $II^\geq_k$?

Key findings

• The set of multiplier sequences of the third kind—those preserving the class $II^\geq_k$ of sign-independently real-rooted polynomials with nonnegative coefficients—is characterized by the system of inequalities $kx_j \leq j(x_k - x_0)$ for $j = 1, \dots, k-1$.
• The recession cone $C_k'$ of the amoeba complement $\Gamma_k'$ is explicitly described by these inequalities, and corresponds exactly to the set of multiplier sequences of the third kind.
• The map $\mathrm{Log}|\cdot|$ is a diffeomorphism from $II^\geq_k$ onto $\Gamma_k'$, establishing a geometric correspondence between the polynomial class and the amoeba domain.
• The necessity of the inequalities is proven by constructing perturbations $p_j^\pm_\varepsilon$ that violate the conditions and lead to non-real-rooted images under $T_\gamma$.
• The sufficiency of the inequalities follows from the fact that any $\gamma$ satisfying them lies in the recession cone $C_k'$, ensuring $T_\gamma(II^\geq_k) \subseteq II^\geq_k$.
• The results generalize classical multiplier sequence theory by introducing a new class of sequences preserved under diagonal operators, with a geometric and algebraic characterization rooted in discriminant amoebae.