arXiv:2606.29125v1

Transcendental Epsilon Multiplicity via Divisor Volumes

Sudipta Das, Stephen Landsittel, Vinh Anh Pham

math.ACmath.AG13H1514C2011J86

Paper metadata

Original submission: June 28, 2026
Claimed MSC: 13H15, 14C20, 11J86
arXiv categories: math.AC, math.AG
Paper license: CC BY-NC-ND 4.0 ↗
Comments: 33 pages

Abstract

We prove that epsilon multiplicity can take transcendental values. The main structural result is a one-ideal formula for section rings: under natural positivity hypotheses, the epsilon multiplicity of an ideal generated in one degree is equal to an integral of a divisor-volume function. This formula transports an asymptotic colength invariant of ideals to the geometry and arithmetic of divisor volumes. To produce a transcendental value, we combine the formula with a shifted projective-bundle construction inspired by Bornträger and Nickel. The shift places the construction in the positivity range required by the one-ideal formula while preserving the underlying disk geometry of the volume computation. Reversing the order of integration reduces the resulting integral to three integrals of rational functions. Their arctangent terms cancel exactly, whereas the remaining real logarithms form an explicit algebraic linear combination whose value is positive. Baker's theorem then implies transcendence. Consequently, there exists a homogeneous ideal in a normal standard graded domain whose epsilon multiplicity is transcendental.

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Generated August 15, 2026

01Statements2 reported findingsCorrect

The one-ideal divisor-volume formula and the construction of a normal five-dimensional standard graded domain with an ideal of transcendental epsilon multiplicity are correct as stated.

Theorem B / Theorem 3.3Correct

Epsilon multiplicity is correctly expressed by the divisor-volume integral

Pages 2 and 11–13 · Theorem B and Theorem 3.3 · arXiv:2606.29125v1

For the ideal generated by $H^0(X,cL-D)$ in degree $c$ , saturation of its $n$ -th power is the divisorial module attached to $nD$ . The regularity hypotheses identify the degree pieces of the actual and divisorial powers above the linear cutoff. Summing the remaining Hilbert-function difference and applying uniform volume asymptotics gives $\varepsilon(JS_+)= (r+1)\int_{\tau_L(D)}^c \operatorname{vol}_X(tL-D)\,dt.$ The endpoint and normalization factors agree with $\dim R(X,L)=r+1$ .

Theorem A / Theorem 6.1Correct

A transcendental epsilon multiplicity is constructed

Pages 2 and 30–31 · Theorem A and Theorem 6.1 · arXiv:2606.29125v1

The shifted projective-bundle construction yields a smooth fourfold with a projectively normal ample polarization, so its section ring is a normal standard graded domain of dimension five. Proposition 4.3 supplies every positivity and regularity hypothesis of Theorem 3.3. The volume computation reduces the resulting epsilon multiplicity to a nonzero algebraic factor times the integral proved transcendental in Lemma 5.4, hence the localized homogeneous ideal has transcendental epsilon multiplicity.

02Proofs4 reported findingsCorrect

The algebraic, geometric, and transcendence arguments supporting the two central results are correct and complete. In particular, the logarithmic term is shown to be nonzero before Baker's theorem is invoked.

Theorem 3.3Correct and complete

Saturation, regularity, and volume asymptotics

Pages 11–13 · Proposition 3.2 and Theorem 3.3 · arXiv:2606.29125v1

Sheafification identifies the saturation with the divisorial module, and the stated multiplication and regularity hypotheses make the comparison with powers of the single degree- $c$ ideal uniform in $n$ . The low-degree contribution is finite length, the Riemann sums have the correct $n^{r+1}$ scale, and continuity and homogeneity of volume justify the limiting integral.

Proposition 4.3 and Proposition 5.2Correct and complete

Shifted projective-bundle geometry and the disk-volume formula

Pages 20–24 · Proposition 4.3 and Proposition 5.2 · arXiv:2606.29125v1

Choosing the common shift and rescaling sufficiently positive makes the required divisors ample and the section ring projectively normal without altering the transverse triangle. The projective-bundle volume formula, the simplex-to-triangle Jacobian, and the circular nef cone give the factor $3N/2$ and the function $t^4\Phi(R(t))$ . Continuity removes the temporary rationality restriction on $t$ .

Lemma 5.4 and Proposition 5.5Correct and complete

Noncancellation and Baker-theorem step

Pages 25–30 · Lemma 5.4 and Proposition 5.5 · arXiv:2606.29125v1

Tonelli's theorem is applicable because the integrand is nonnegative. Splitting the polar integral along the three edge chambers produces rational terms, real logarithms of positive algebraic numbers, and arctangent terms that cancel in the full sum. The manuscript separately proves that the logarithmic part is positive. After reduction to a maximal linearly independent family of logarithms, Baker's theorem therefore makes the value transcendental; no independence of the original displayed logarithms is assumed.

Theorem 6.1Correct and complete

Final transfer to epsilon multiplicity

Pages 30–31 · proof of Theorem 6.1 · arXiv:2606.29125v1

The dimension shift from the fourfold to its section ring is handled correctly, the ideal is generated in the declared degree, and Theorem 3.3 applies with $c=q$ . The resulting integral is exactly the one evaluated in Section 5, and multiplication by its positive algebraic prefactor preserves transcendence.

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No non-novelty findings.

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Paper: arXiv:2606.29125v1
Authors listed: Sudipta Das, Stephen Landsittel, Vinh Anh Pham
Audit date: August 15, 2026

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Transcendental Epsilon Multiplicity via Divisor Volumes

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Epsilon multiplicity is correctly expressed by the divisor-volume integral

A transcendental epsilon multiplicity is constructed

Saturation, regularity, and volume asymptotics

Shifted projective-bundle geometry and the disk-volume formula

Noncancellation and Baker-theorem step

Final transfer to epsilon multiplicity

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