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arxiv: 2604.02453 · v1 · submitted 2026-04-02 · 🧮 math.CV · math.PR

Zeros of random P-polynomials in mathbb{C}^d with exponential profiles

Turgay Bayraktar , Afrim Bojnik This is my paper

Pith reviewed 2026-05-13 20:08 UTC · model grok-4.3

classification 🧮 math.CV math.PR
keywords random polynomialszero currentsplurisubharmonic functionsLegendre transformconvex bodiestoric geometrymultivariate zeros
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The pith

Normalized zero currents of random P-polynomials in C^d converge weakly to dd^c of a deterministic toric plurisubharmonic function given by the Legendre transform of the profile over P.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that random multivariate polynomials whose monomial supports are taken from nP for a fixed convex body P and whose coefficients obey a uniform exponential profile f have normalized potentials converging in probability to a deterministic limit. This limit is the toric plurisubharmonic function whose complex Hessian is the limiting zero current. The convergence holds under a power-law tail condition on the coefficient magnitudes and strengthens to almost sure convergence along suitable sequences when a d-th logarithmic moment exists. The limiting object on the torus is expressed directly via the Legendre-Fenchel transform of f over P, linking random zero hypersurfaces to convex-analytic data.

Core claim

Under the tail condition P(log(1+|ξ0|)>t)=o(t^{-d}), the normalized potentials (1/n)log|P_n| converge in probability in L^1_loc(C^d) to Φ_{P,f}, so that the normalized zero currents (1/n)[Z_{P_n}] converge weakly to the positive closed (1,1)-current dd^c Φ_{P,f}. On (C*)^d the potential equals the Legendre-Fenchel transform I_{P,f} of the profile evaluated at Log z. Stronger moment assumptions yield almost sure convergence of the currents along the full sequence when d>2 and along sparse subsequences when d≤2.

What carries the argument

The deterministic toric plurisubharmonic function Φ_{P,f} defined as the Legendre-Fenchel transform I_{P,f} of the exponential profile f over the convex body P; its complex Hessian dd^c Φ_{P,f} is the limiting zero current.

If this is right

  • The asymptotic support of the zeros is completely determined by the pair (P,f) through the zero set of dd^c Φ_{P,f}.
  • The limiting distribution is insensitive to the precise law of the coefficients beyond the exponential profile f and the stated tail condition.
  • Almost sure convergence of the currents holds for d>2 under the single extra assumption that E[(log(1+|ξ0|))^d]<∞.
  • The results recover and extend the one-variable exponential-profile theorems of Kabluchko-Zaporozhets to the genuinely multivariate toric setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the tail condition is violated, zeros may escape to infinity with positive probability and the convergence statements fail.
  • The same Legendre-transform mechanism may govern zeros of random sections of toric line bundles on projective toric varieties.
  • Explicit computation of Φ_{P,f} for polytopes P with few vertices would give concrete, testable formulas for the limiting zero distribution in low dimensions.

Load-bearing premise

The i.i.d. complex coefficients satisfy the tail bound P(log(1+|ξ0|)>t)=o(t^{-d}).

What would settle it

A sequence of coefficient realizations obeying the tail condition for which (1/n)log|P_n| fails to converge in L^1_loc to Φ_{P,f} or for which the zero currents (1/n)[Z_{P_n}] do not converge weakly to dd^c Φ_{P,f}.

read the original abstract

We study random multivariate $P$-polynomials in $\mathbb{C}^d$ with monomial supports constrained to $nP\cap\mathbb{Z}_+^d$ for a convex body $P\subset\mathbb{R}_+^d$, and deterministic coefficients admitting a uniform exponential profile $f$ on $P$. Assuming the tail condition $\mathbb{P}(\log(1+|\xi_0|)>t)=o(t^{-d})$ on the i.i.d. complex coefficients, we prove that the normalized potentials $\frac1n\log|\mathbf{P}_n|$ converge in probability in $L^1_{\mathrm{loc}}(\mathbb{C}^d)$ to a deterministic toric plurisubharmonic function $\Phi_{P,f}$, and consequently the normalized zero currents $\frac1n[Z_{\mathbf{P}_n}]$ converge weakly to the closed positive $(1,1)$-current $dd^c\Phi_{P,f}$. Under the stronger logarithmic moment assumption $\mathbb{E}[(\log(1+|\xi_0|))^d]<\infty$, we prove almost sure weak convergence of the zero currents along the full sequence for $d>2$, and along sparse subsequences for $d \le 2$. On $(\mathbb{C}^*)^d$, the limiting potential is given by $\Phi_{P,f}(z)=I_{P,f}(\operatorname{Log} z)$, where $I_{P,f}$ is the Legendre-Fenchel transform of the profile over $P$ and $\operatorname{Log} (z)=(\log|z_1|,\dots,\log|z_d|)$. These results extend the exponential-profile mechanism of Kabluchko and Zaporozhets from one complex variable to the genuinely multivariate $P$-polynomial setting under relaxed probabilistic assumptions, directly connecting random zero hypersurfaces with convex-analytic data determined by $(P,f)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper studies random multivariate P-polynomials in C^d whose monomial supports lie in nP ∩ Z_+^d for a fixed convex body P ⊂ R_+^d, with i.i.d. complex coefficients admitting a deterministic exponential profile f on P. Under the tail condition P(log(1+|ξ_0|)>t)=o(t^{-d}), it proves that (1/n)log|P_n| converges in probability in L^1_loc(C^d) to a deterministic toric plurisubharmonic function Φ_{P,f}; consequently the normalized zero currents (1/n)[Z_{P_n}] converge weakly to the positive closed (1,1)-current dd^c Φ_{P,f}. On (C*)^d the limit is given explicitly by the Legendre-Fenchel transform I_{P,f}(Log z). Under the stronger assumption E[(log(1+|ξ_0|))^d]<∞, almost-sure weak convergence of the currents holds along the full sequence when d>2 and along sparse subsequences when d≤2. The work extends the Kabluchko-Zaporozhets exponential-profile mechanism from one variable to the genuinely multivariate toric setting.

Significance. If the proofs hold, the results furnish a clean multivariate extension that directly ties the asymptotic zero distribution of random polynomials with constrained supports to convex-analytic data via the Legendre transform. The relaxed tail hypothesis is a genuine improvement over stronger moment assumptions in the literature, and the reduction to toric pluripotential theory is natural. The explicit identification of the limiting current with dd^c I_{P,f} supplies a falsifiable prediction that can be checked numerically for concrete (P,f).

major comments (2)
  1. [§3] §3 (convergence in probability): the union-bound argument that the tail condition implies P(max_α log|ξ_α| > n s) → 0 for every s>0 is only sketched; an explicit estimate showing that the O(n^d) monomials produce an o(n) random contribution uniformly on compact subsets of C^d is needed to justify the subsequent L^1_loc comparison with the toric psh envelope.
  2. [Theorem 1.3] Theorem 1.3 (a.s. convergence): for d≤2 the Borel-Cantelli argument is applied along a sparse subsequence; the manuscript should state the precise density of this subsequence and verify that the weak limit of the currents is nevertheless independent of the subsequence choice.
minor comments (3)
  1. [Abstract and §1] The notation P_n versus bold P_n is used inconsistently in the abstract and §1; adopt a single convention.
  2. [§2] The definition of the toric psh envelope and the precise domain of the Legendre-Fenchel transform I_{P,f} should be recalled in §2 before the main statements.
  3. [Theorem 1.1] Add a short remark after Theorem 1.1 clarifying that the limiting current is independent of the choice of local coordinates on (C*)^d.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will incorporate the requested clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (convergence in probability): the union-bound argument that the tail condition implies P(max_α log|ξ_α| > n s) → 0 for every s>0 is only sketched; an explicit estimate showing that the O(n^d) monomials produce an o(n) random contribution uniformly on compact subsets of C^d is needed to justify the subsequent L^1_loc comparison with the toric psh envelope.

    Authors: We agree that the union-bound argument in §3 should be made fully explicit. In the revision we will insert the following estimate immediately after the sketch: Let M_n = max_{α ∈ nP ∩ ℤ_+^d} log|ξ_α|. By the union bound, P(M_n > n s) ≤ N_n ⋅ P(log(1+|ξ_0|) > n s), where N_n = #(nP ∩ ℤ_+^d) = O(n^d). The given tail condition yields P(log(1+|ξ_0|) > t) = o(t^{-d}), so P(M_n > n s) = O(n^d) ⋅ o((n s)^{-d}) = o(1) for every fixed s > 0. On any compact K ⊂ ℂ^d we have log|z^α| ≤ n C_K uniformly in α, hence the maximal random term is o(n) with high probability uniformly on K. This justifies the subsequent L^1_loc comparison with the toric psh envelope. We will add this calculation verbatim. revision: yes

  2. Referee: [Theorem 1.3] Theorem 1.3 (a.s. convergence): for d≤2 the Borel-Cantelli argument is applied along a sparse subsequence; the manuscript should state the precise density of this subsequence and verify that the weak limit of the currents is nevertheless independent of the subsequence choice.

    Authors: We will revise the statement of Theorem 1.3 and the proof to specify the subsequence explicitly. For d ≤ 2 we take n_k = ⌈k^2⌉ (any subsequence with n_{k+1}/n_k → ∞ and zero asymptotic density works). Under the assumption E[(log(1+|ξ_0|))^d] < ∞ the probabilities of the exceptional sets for the events at n_k are O(k^{-2-ε}) for some ε > 0, hence summable, and Borel-Cantelli yields almost-sure convergence along {n_k}. Independence of the subsequence follows from the convergence in probability of the full sequence established in Theorem 1.2: any subsequence limit must coincide with the unique in-probability limit dd^c Φ_{P,f}. We will add a short clarifying paragraph after the proof of Theorem 1.3 stating the density (zero) and this uniqueness argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; limiting object defined independently

full rationale

The paper defines the target potential Φ_{P,f} explicitly as the Legendre-Fenchel transform I_{P,f} of the given deterministic profile f over the convex body P; this construction uses only convex analysis and does not depend on the random coefficients or their realizations. The tail condition P(log(1+|ξ_0|)>t)=o(t^{-d}) is used solely to obtain a uniform probabilistic bound showing that max_α log|ξ_α| = o(n) with high probability, after which standard comparison with the toric psh envelope and weak continuity of dd^c yield the L^1_loc convergence of (1/n)log|P_n| and the current convergence. These steps invoke only classical pluripotential theory and Borel-Cantelli arguments; no parameter is fitted to the zero data, no self-citation supplies a uniqueness theorem that forces the result, and the cited Kabluchko-Zaporozhets mechanism is external prior work. Consequently the derivation chain does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard background from pluripotential theory and convex analysis; no free parameters are fitted to data and no new entities are postulated.

axioms (2)
  • standard math Existence and basic properties of the Legendre-Fenchel transform for convex bodies and profiles
    Invoked to define the limiting potential Φ_{P,f} on (C*)^d
  • standard math Weak convergence of currents and L1_loc convergence of potentials imply convergence of zero sets
    Standard in pluripotential theory, used to pass from potential convergence to zero-current convergence

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Reference graph

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