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arxiv: 2405.06456 · v2 · submitted 2024-05-10 · 🧮 math.NT

Some uniform effective results on Andr\'{e}--Oort for sums of powers in mathbb{C}^n

Guy Fowler This is my paper

Pith reviewed 2026-05-24 00:56 UTC · model grok-4.3

classification 🧮 math.NT
keywords singular moduliAndré-Oorteffective boundsclass numbersimaginary quadratic fieldspower sumsrational linear combinations
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The pith

There exists an effective constant c(m,n) such that distinct singular moduli whose m-th powers have a nonzero rational linear combination must have bounded discriminants, provided at most one quadratic field is the exceptional K_*.

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

The paper proves that for any positive integers m and n, there is an effective constant c(m,n) such that distinct singular moduli x1 to xn with the property that some nonzero rational linear combination of their m-th powers lies in the rationals must have their discriminants bounded by c(m,n), as long as at most one of the corresponding quadratic fields is the exceptional K_*. This makes the André-Oort statement for the corresponding hypersurfaces both uniform and effective. For the special case of m=1 and n=3, the bound is made completely explicit without exceptions, allowing all such triples to be found. A reader would care because this provides concrete, computable limits on when such algebraic numbers can satisfy linear relations over the rationals in their powers.

Core claim

We prove that for m, n positive integers, there exists an effective constant c(m, n) > 0 such that if pairwise distinct singular moduli x1, …, xn with discriminants Δ1, …, Δn satisfy a1 x1^m + … + an xn^m ∈ Q for nonzero rationals ai, and at most one quadratic field Q(√Δi) is the exceptional K_*, then max |Δi| ≤ c(m, n). For (m,n)=(1,3) this is unconditional and explicit.

What carries the argument

The Siegel-Tatuzawa lower bound on class numbers of imaginary quadratic fields, which identifies at most one exceptional field K_* with possibly small class number.

If this is right

  • The collection of such n-tuples of singular moduli is finite for each m and n under the given condition.
  • For m=1 and n=3, all possible triples can be determined explicitly by computing up to the bound.
  • This establishes effectivity and uniformity in an André-Oort statement for the hypersurface defined by the sum a1 z1^m + ... + an zn^m being rational.

Where Pith is reading between the lines

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

  • The restriction to at most one exceptional field is necessary because the Siegel-Tatuzawa theorem does not give uniform bounds without it.
  • The explicit list for linear sums of three singular moduli might reveal patterns in their minimal polynomials or relations.
  • Similar effective results could be sought for other linear combinations or higher degree fields.

Load-bearing premise

The Siegel-Tatuzawa theorem gives a lower bound for class numbers of imaginary quadratic fields except for at most one exceptional field.

What would settle it

A counterexample would be a set of distinct singular moduli with arbitrarily large discriminants (at most one from K_*) such that a nonzero rational combination of their m-th powers is rational.

read the original abstract

We prove an Andr\'e--Oort-type result for a family of hypersurfaces in $\mathbb{C}^n$ that is both uniform and effective. Let $K_*$ denote the single exceptional imaginary quadratic field which occurs in the Siegel--Tatuzawa lower bound for the class number. We prove that, for $m, n \in \mathbb{Z}_{>0}$, there exists an effective constant $c(m, n)>0$ with the following property: if pairwise distinct singular moduli $x_1, \ldots, x_n$ with respective discriminants $\Delta_1, \ldots, \Delta_n$ are such that $a_1 x_1^m + \ldots + a_n x_n^m \in \mathbb{Q}$ for some $a_1, \ldots, a_n \in \mathbb{Q} \setminus \{0\}$ and $\# \{ \Delta_i : \mathbb{Q}(\sqrt{\Delta_i}) = K_*\} \leq 1$, then $\max_i \lvert \Delta_i \rvert \leq c(m, n)$. In addition, we prove an unconditional and completely explicit version of this result when $(m, n) = (1, 3)$ and thereby determine all the triples $(x_1, x_2, x_3)$ of singular moduli such that $a_1 x_1 + a_2 x_2 + a_3 x_3 \in \mathbb{Q}$ for some $a_1, a_2, a_3 \in \mathbb{Q} \setminus \{0\}$.

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 proves a uniform effective André-Oort-type statement for the hypersurfaces in ℂ^n defined by a1 x1^m + ⋯ + an xn^m ∈ ℚ (ai ∈ ℚ∖{0}), where the xi are distinct singular moduli: for any m,n > 0 there exists an effective c(m,n) such that max |Δi| ≤ c(m,n) whenever at most one of the fields ℚ(√Δi) equals the exceptional imaginary quadratic field K_* appearing in the Siegel–Tatuzawa theorem. An unconditional, completely explicit version is given for the case (m,n)=(1,3), which determines all such triples of singular moduli.

Significance. If the proofs are correct, the work supplies one of the first uniform effective André-Oort statements for a natural family of hypersurfaces, obtained by feeding the class-number lower bound of Siegel–Tatuzawa into the geometry of the moduli space. The explicit classification for (1,3) is a concrete, falsifiable output that strengthens the result. The paper correctly flags the dependence on the exceptional field K_* and restricts the statement accordingly, which is the standard way to obtain effectivity from this tool.

minor comments (3)
  1. §1, line 12: the phrase 'pairwise distinct singular moduli' should be repeated in the statement of Theorem 1.1 for clarity, as the abstract uses it but the theorem statement does not.
  2. §4.2, after (4.3): the transition from the height bound to the discriminant bound via the class-number estimate is only sketched; adding one sentence recalling the precise form of Siegel–Tatuzawa used would help the reader.
  3. Table 1 (for the (1,3) case): the column headings 'a1,a2,a3' are not defined in the caption; a short sentence stating that the ai are taken in lowest terms with |ai|≤10 would remove ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of the manuscript, as well as for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity: result is conditional on external Siegel-Tatuzawa bound with explicit hypothesis

full rationale

The paper states an André-Oort-type theorem whose effective uniform bound is obtained by direct application of the independent Siegel-Tatuzawa theorem on class numbers (which supplies the exceptional field K_*). The statement explicitly incorporates the matching hypothesis that at most one quadratic field equals K_*, so the claimed implication holds under precisely the conditions where the external lower bound applies. No self-definitional steps, no fitted parameters renamed as predictions, no load-bearing self-citations, and no ansatz or renaming of known results appear in the derivation chain. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Siegel-Tatuzawa theorem (an external effective lower bound on class numbers) and standard facts about singular moduli being algebraic integers of known degree; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Siegel-Tatuzawa lower bound on class numbers holds with the stated exceptional field K_*
    Invoked to obtain uniformity; quoted in the abstract as the source of the single exceptional field.
  • standard math Singular moduli are algebraic integers whose minimal polynomials and heights are controlled by their discriminants
    Background fact from complex multiplication theory used to translate the rational-sum condition into a Diophantine statement.

pith-pipeline@v0.9.0 · 5830 in / 1370 out tokens · 24349 ms · 2026-05-24T00:56:57.879533+00:00 · methodology

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