Inequalities involving polynomials and quasimodular forms

Abstract: In this paper, we study inequalities involving polynomials and quasimodular forms. More precisely, we focus on the monotonicity of the functions of the form $t \mapsto t^m F(it)$ where $F$ is a quasimodular form and $m > 0$. As an application, we construct infinitely many positive quasimodular forms of level $> 1$. We also give alternative proofs of modular form inequalities used in the proof of optimality of Leech lattice packing and universal optimality of the lattice by Cohn, Kumar, Miller, Radchenko, and Viazovska.

• The paper establishes an algebraic framework for proving the monotonicity of t^m F(it) and related inequalities.
• It details sufficient conditions and recurrence relations to construct positive quasimodular forms at higher levels.
• The work offers alternative algebraic proofs of modular form inequalities, impacting sphere packing and universal optimality results.

Summary of "Inequalities involving polynomials and quasimodular forms" (2602.10536)

Background and Motivation

The paper addresses inequalities involving polynomials and quasimodular forms, particularly those that arise in the context of optimal sphere packing and universal optimality for the $E_8$ and Leech lattices. The construction of "magic functions" in the Cohn--Elkies linear programming bound, leveraging quasimodular forms, has been pivotal in proving these results. Viazovska’s solution of the sphere packing problem in dimension 8 and subsequent results for dimension 24 by Cohn et al. critically depend on positivity properties of quasimodular forms and associated inequalities. The original proofs often rely on computer-assisted rigorous numerics; the present paper aims to establish a more conceptual, algebraic framework for such inequalities and their monotonicity properties.

Framework and Key Results

The central object of study is the function $t \mapsto t^m F(it)$ for a quasimodular form $F$ and parameter $m > 0$. The monotonicity of these functions is linked to the positivity of certain combinations involving $F(it)$ and its derivative, specifically:

$t \mapsto - F(it) + \frac{2\pi t}{m} F'(it),$

and to Rankin–Cohen brackets. The paper advances a general framework for proving monotonicity and related inequalities, subsuming as special cases various modular form inequalities previously employed in sphere packing and universal optimality arguments.

Sufficient Conditions for Monotonicity

Two algebraic sufficient conditions are provided:

• If $F$ and $F'$ are positive, the tangent line at $t=0$ for $\frac{F(it)}{F'(it)}$ must have slope $\frac{2\pi}{m}$, and a corresponding Rankin–Cohen bracket $(m+1)(F')^2 - m F'' F$ must be positive, then $t \mapsto t^m F(it)$ is monotone decreasing.
• Monotonicity of $t^{m+1} F'(it)$ implies monotonicity of $t^m F(it)$ when $F$ is a cusp form.

Applications

Construction of Positive Quasimodular Forms at Higher Levels

Using monotonicity, the paper constructs an infinite family of positive quasimodular forms of level $>1$, which typically are not completely positive (i.e., their Fourier coefficients are not all positive). Explicit examples for levels 2 and 4 are provided, together with precise density computations for sign distributions of Fourier coefficients.

Algebraic Proofs of Modular Form Inequalities

Alternative algebraic proofs (as opposed to numerically assisted) are given for key inequalities, previously used in the proof of the Leech lattice packing and its universal optimality, aligning with recent trends towards conceptual proofs in the domain. The monotonicity results underpin these arguments:

• The modular form inequality for the Leech lattice (cf. [cohn2017sphere]) is reduced to showing positivity of a quasimodular form, which is factored and then further reduced to linear combinations of explicitly positive quasimodular forms.
• The universal optimality inequality (cf. [cohn2022universal]) is shown to be equivalent to monotonicity of $t \mapsto t^{11} X_{12,1}(it)$ for the extremal quasimodular form $X_{12,1}$, with monotonicity proved by algebraic criteria.

Monotonicity and Extremal Quasimodular Forms

The monotonicity for exponents $m = w - 1$ for extremal depth 1 quasimodular forms $X_{w,1}$ is conjectured (excluding $w=8,10$), and weaker results are established in related recurrences. Explicit algebraic recurrence relations—including for constant terms and derivatives—are developed for extremal quasimodular forms.

Numerical Results and Contradictory Claims

Strong numerical results include monotonicity verification for numerous forms, optimal bounds for exponents yielding positivity, and density bounds for coefficients. Contradictory claims emerge naturally: not all powers yield monotonicity (e.g., $t^7 X_{8,1}(it)$ and $t^9 X_{10,1}(it)$ are not monotone), and positivity does not guarantee complete positivity.

Implications and Future Developments

Theoretical implications include new algebraic tools for modular and quasimodular forms in lattice packing and universal optimality, generalization to non-level-1 situations, and refined understanding of positivity and monotonicity. Practically, the results enable more efficient and conceptual proofs, circumventing full reliance on numerics, and suggest potential for automated verification using formal systems (Lean and Sage codes are referenced).

The density conjecture regarding positive coefficients in positive quasimodular forms, as well as the conjecture on complete positivity for certain explicit families, pose concrete future directions. The paper further speculates that monotonicity criteria may be extendable to other universality inequalities and modular optimization contexts.

Conclusion

This work establishes a detailed algebraic framework for inequalities and monotonicity in polynomials and quasimodular forms, provides constructive criteria, and applies them to major problems in modular form theory and sphere packing. It bridges gaps in conceptual understanding, delivers new positive quasimodular forms of higher levels, and sets forth conjectures for coefficient density and complete positivity. Future prospects include resolving the density conjecture, extending algebraic methods to additional modular inequalities, and leveraging formal verification tools for advanced modular form analysis.