Inequalities involving polynomials and quasimodular forms

TL;DR

This work develops a general framework to analyze inequalities for quasimodular forms through the monotonicity of $t^m F(it)$, linking positivity properties to level-raising and to modular-form inequalities underlying lattice packing results. It introduces two practical monotonicity criteria, studies extremal quasimodular forms $X_{w,1}$ via recurrences and decompositions, and derives numerous concrete applications, including construction of positive forms of higher level and algebraic proofs of Leech lattice optimality results. By proving new positivity results and proposing conjectures about density of positive coefficients and complete positivity at higher levels, the paper provides conceptual tools for positivity and monotonicity phenomena in the theory of quasimodular forms with potential impact on sphere packing, potential theory, and modular-inequality proofs. Overall, the approach bridges analytic, algebraic, and combinatorial aspects of quasimodular forms to yield new proofs and constructions with implications for extremal lattices and related inequalities. The work also raises open questions about the natural density of positive Fourier coefficients and the extent of complete positivity across levels and weights.

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.

Figures (4)

• Figure 1: Graphs of $g_8(t)$ and $g_9(t)$ on $2.5 \le t \le 20$.
• Figure 2: Graphs of $t \mapsto t^{7} X_{8, 1}(it)$ and $t \mapsto X_{8, 1}(it) / X_{8, 1}'(it)$ with asymptotes as $t \to 0^+$.
• Figure 3: Graphs of $t^8 X_{10, 1}(it)$ and $t^9 X_{10, 1}(it)$ on $0.1 \le t \le 5$.
• Figure 4: Graphs of $t \mapsto t^{11} X_{12, 1}(it)$ and $t \mapsto X_{12, 1}(it) / X_{12, 1}'(it)$ with asymptotes as $t \to 0^+$ and $t \to \infty$.

Theorems & Definitions (58)

• Lemma 2.1
• proof
• Definition 2.2: L. lee2024algebraic
• Proposition 2.3: L. lee2024algebraic
• Theorem 2.4: Grabner grabner2020quasimodular
• Theorem 2.5: L. lee2024algebraic
• Proposition 2.6
• proof
• Lemma 2.7
• proof