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arxiv: 2606.10523 · v1 · pith:HPYF3GSLnew · submitted 2026-06-09 · 🧮 math.NT

On the Ekedahl sieve for the singular locus of the discriminant polynomial

Gaurav Digambar Patil This is my paper

Pith reviewed 2026-06-27 11:56 UTC · model grok-4.3

classification 🧮 math.NT
keywords Ekedahl sievediscriminant polynomialsingular locussquarefree valuesnumber field enumerationsieve methodsarithmetic enumeration
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The pith

The Ekedahl sieve for the discriminant singular locus bypasses inductive steps by exploiting non-degeneracy outside the two extreme coefficients.

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

This paper develops a specialized variant of the Ekedahl sieve for the singular locus of the discriminant polynomial. It exploits non-degeneracy properties away from the two most extreme coefficients to avoid the standard inductive framework, reducing the process to one or two steps in many cases. The approach delivers robust tail-end estimates along with squarefree power-saving bounds that incorporate external modular conditions. These changes increase the allowable size of the sieving modulus and improve error terms when counting bounded squarefree polynomial values, while supplying the geometric estimates needed for weighted counts of number fields by discriminant.

Core claim

By exploiting the specific non-degeneracy properties of the discriminant outside its two most extreme coefficients, we bypass the standard inductive framework, reducing the sieve to a highly efficient two-step process in some cases, and a one step process in others. We establish robust generic tail-end estimates as well as squarefree, power-saving bounds that seamlessly incorporate external modular conditions.

What carries the argument

Specialized Ekedahl sieve variant for the singular locus of the discriminant polynomial, which uses non-degeneracy outside extreme coefficients to eliminate inductive steps.

Load-bearing premise

The discriminant polynomial possesses specific non-degeneracy properties outside its two most extreme coefficients that permit bypassing the standard inductive sieve framework entirely.

What would settle it

A concrete computation for a binary form discriminant showing that the non-degeneracy property fails for some tail variable, forcing extra inductive steps and producing worse than claimed error terms.

read the original abstract

The Ekedahl sieve is a powerful tool for enumerating arithmetic objects, but traditional formulations relying on inductive steps often yield suboptimal bounds when applied to highly skew boxes. This limitation is particularly restrictive when introducing large modular conditions that compete with the tail-end variables of a binary form. In this paper, we develop a specialized variant of the Ekedahl sieve tailored to the singular locus of the discriminant polynomial. By exploiting the specific non-degeneracy properties of the discriminant outside its two most extreme coefficients, we bypass the standard inductive framework, reducing the sieve to a highly efficient two-step process in some cases, and a one step process in others. We establish robust generic tail-end estimates as well as squarefree, power-saving bounds that seamlessly incorporate external modular conditions. This optimization maximizes the permissible range of the sieving modulus, yielding improved error terms for the enumeration of bounded squarefree values of certain polynomials and providing the foundational geometric sieve estimates required for the weighted enumeration of number fields by discriminant.

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

0 major / 3 minor

Summary. The manuscript develops a specialized variant of the Ekedahl sieve for the singular locus of the discriminant polynomial. By exploiting non-degeneracy properties of the discriminant outside its two most extreme coefficients, the authors bypass the standard inductive framework and reduce the sieve to a one-step or two-step process in various cases. They establish generic tail-end estimates together with squarefree power-saving bounds that incorporate external modular conditions, with the goal of improving error terms for counting bounded squarefree values of polynomials and supplying foundational estimates for the weighted enumeration of number fields by discriminant.

Significance. If the non-degeneracy properties are shown to suffice for the claimed reduction while producing the stated tail-end and power-saving bounds, the work would enlarge the permissible range of the sieving modulus in arithmetic enumeration problems involving discriminants. This could yield sharper error terms in the study of squarefree values and number-field counting. The explicit handling of large modular conditions competing with tail variables is a potentially useful technical advance.

minor comments (3)
  1. [§1] §1, paragraph 3: the phrase 'robust generic tail-end estimates' is used without an immediate pointer to the precise statement (e.g., Theorem 3.4 or Proposition 4.2) that encodes the improvement over the classical inductive bound.
  2. [§2.3] §2.3: the definition of the singular locus S(D) could include an explicit local equation or ideal membership condition to make the subsequent non-degeneracy statements easier to verify.
  3. [Notation] Notation list: the symbols E_p and u_p appearing in the tail-end estimates are introduced only in the proof of Lemma 5.1; a forward reference or consolidated notation table would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and the recommendation for minor revision. The report provides a helpful summary of the work but does not list any specific major comments requiring point-by-point response.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external non-degeneracy properties

full rationale

The paper presents a specialized Ekedahl sieve variant that bypasses inductive steps by exploiting non-degeneracy properties of the discriminant outside extreme coefficients. No equations, definitions, or claims in the provided abstract reduce a result to its own inputs by construction, nor do they rely on self-citation chains or fitted parameters renamed as predictions. The central claims are framed as consequences of stated geometric properties of the polynomial, which are treated as independent inputs rather than derived within the paper. This is the expected non-circular outcome for a methods paper introducing a tailored sieve.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review is based solely on the abstract; no explicit free parameters, invented entities, or ad-hoc axioms are stated. The work assumes standard background on the discriminant polynomial and sieve methods.

axioms (1)
  • domain assumption The discriminant polynomial has non-degeneracy properties outside its two most extreme coefficients.
    This property is invoked to justify bypassing the inductive framework.

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Reference graph

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