Algebraic independence of certain entire functions of two variables generated by linear recurrences

Haruki Ide

REVIEW 3 major objections 4 minor 13 references

Algebraic independence of certain entire functions of two variables generated by linear recurrences

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper constructs an explicit two-variable entire function whose values and all partial derivatives at algebraic points are algebraically independent, uniformly over complex and p-adic fields.

desk verdict The construction of Theta and the reduction of Theorem 5 to Theorem 6 are genuinely clever, but Lemma 6's assertion that the evaluation point (1,...,1,a) satisfies condition (III)_p is false, so the proof of the main theorem does not go through as written. read the letter →

arxiv 1908.06289 v1 pith:Z6ICEXGW submitted 2019-08-17 math.NT

Haruki Ide This is my paper · ORCID

classification math.NT MSC 11J8511B37

keywords algebraicindependenceentirefunctionsMahlerlinearrecurrencesp-adictranscendentalnumbertheorypartialderivativesinfiniteproductstranscendence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper constructs an explicit entire function of two variables whose values and all partial derivatives at every algebraic point with nonzero first coordinate are algebraically independent over the rationals. The function is built from a linear recurrence $R_k$ of nonnegative integers and a fixed algebraic number $a$ with $0<|a|_p<1$: $\Theta(x,y)=\sum_{k\ge0}a^{R_k}x^k\prod_{j\ne k}(1-a^{R_j}y)$. Under a spectral condition $(\mathrm{R})_p$ on the recurrence, Theorem 5 asserts that the infinite set of mixed partial derivatives at algebraic points $(\alpha,\beta)$ with $\alpha\ne0$, together with the derivatives $G^{(N_\beta)}(\beta)$ of the infinite product $G(y)=\prod_k(1-a^{R_k}y)$, is algebraically independent, uniformly for $p=\infty$ and for every prime $p$. This matters because it raises one-variable independence results for the series $F(x)=\Theta(x,0)$ and the product $G(y)$ to a single two-variable object whose mixed partials are also independent, and because the same series works in every completion of $\mathbb{Q}$. The route is to shift the recurrence and re-express the derivative values as values of Mahler functions of several variables; the paper's new criterion reduces their independence to the absence of rational-function solutions of two explicit functional equations.

What carries the argument

The engine is the companion matrix $\Omega$ of the recurrence, acting on monomials $M(\mathbf{z})=z_1^{R_{n-1}}\cdots z_n^{R_0}$ so that $M(\Omega^k\mathbf{z})=z_1^{R_{k+n-1}}\cdots z_n^{R_k}$. The point $\gamma=(1,\ldots,1,a)$ then lies in the domain of convergence, and the values to be shown independent become values of Mahler functions of several variables, satisfying functional equations such as $$ h_{jm}(x;\mathbf{z})=x\,h_{jm}(x;\$\Omega$\mathbf{z})+\left(\frac{M(\mathbf{z})}{1-\beta_j M(\mathbf{z})}\right)^{m+1}. $$ Theorem 7 is the paper's criterion: under four conditions on $\Omega$ and $\gamma$ (a non-root-of-unity spectral condition, growth conditions, and a vanishing condition), algebraic dependence of the values forces either a rational-function solution to a linear functional equation with a repeated eigenvalue or a rational-function solution to a multiplicative equation $S(\mathbf{z})=S(\Omega\mathbf{z})\prod_j(1-\beta_j M(\mathbf{z}))^{d_j}$. Lemma 6 verifies the four conditions in both complex and p-adic cases, and two lemmas from earlier work on recurrences rule out both rational-function alternatives whenever $(\mathrm{R})_p$ holds, giving the contradiction.

What would settle it

Exhibit a nonzero convergent power series $f$ over $\mathbb{C}_p$ and a recurrence satisfying $(\mathrm{R})_p$ such that $f(\Omega^k(1,\ldots,1,a))=0$ for all sufficiently large $k$; the paper's p-adic verification uses Lemma 2 to exclude exactly this, so such a series would disprove the p-adic case of Theorem 5.

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Extended reading notes

Core claim

The central claim, stated as Theorem 5, is that for $p=\infty$ or a prime and any linear recurrence $R_k$ satisfying $(\mathrm{R})_p$, the set $$ \left\{\frac{\$partial^{{l+m}}$\Theta}{\partial x^l\partial y^m}(\$\alpha$,\$\beta$): \$\alpha$\in\bar{\mathbb{Q}}^{\times},\ \$\beta$\in\bar{\mathbb{Q}},\ l\ge0,\ m\ge N_\$\beta$\right\} \cup \{$G^{{(N_\beta)}}$(\$\beta$):\$\beta$\in\bar{\mathbb{Q}}^{\times}\} $$ is algebraically independent over $\mathbb{Q}$ in $\overline{\mathbb{Q}}_p$. Here $G(y)=\prod_k(1-a^{R_k}y)$ and $N_\beta=\operatorname{ord}_{y=\beta}G(y)$. This is an explicit two-variable entire function with the property that one function, together with every mixed partial derivative, produces a new algebraically independent number at each algebraic point with nonzero first coordinate. The proof does not attack the values directly; it proves a stronger statement (Theorem 6) for the auxiliary series $H(x,y)=\sum_k a^{R_k}x^k/(1-a^{R_k}y)$, and then shuttles between $\Theta$ and $H,G$ through triangular invertible changes of coordinates, shifting the recurrence so the evaluation points avoid the zeros of $G$.

Load-bearing premise

The load-bearing premise is that a p-adic vanishing theorem quoted from earlier work (Lemma 2) holds, namely that no nonzero convergent power series can vanish at every iterate of the multiplicative transformation generated by the recurrence, since if that lemma fails the p-adic case of Theorem 5 does not follow.

Editorial extensions

If this is right

Corollary 1: the values $F^{(l)}(\alpha)$ from the power series $F(x)=\Theta(x,0)$ and the derivatives $G^{(m)}(\beta)$ with $m\ge N_\beta$ from the infinite product are jointly algebraically independent, so the one-variable theorems for $F$ and $G$ are refined and unified.
Corollary 2: if in addition the recurrence is strictly increasing, the derivative $\Xi=\partial\Theta/\partial y$ by itself has the same property: its values and all partial derivatives at every algebraic point with $\alpha\ne0$ are algebraically independent; the Fibonacci numbers give an explicit instance.
The construction is uniform in $p$: the same series $\Theta$ works for the complex field and for every p-adic completion, so the algebraic-independence statement holds simultaneously in all these settings.
Because $\Theta(x,0)=F(x)$ and $\Theta(1,y)=-G'(y)$, the mixed partials interpolate between the known one-variable results, which is what the triangular-change-of-variables proof exploits.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The triangular shift-and-replace step that avoids the zeros of $G$ is the kind of argument that should extend to three or more variables: a higher-dimensional recurrence and a larger companion matrix would reduce the analogous independence statement to the same functional-equation dichotomy, assuming the vanishing theorem used here generalizes.
Because the p-adic half rests on Lemma 2, which is quoted rather than proved, the theorem as stated is conditional in the p-adic case; a direct proof of that lemma for the companion matrix would complete the p-adic case, and a counterexample would leave only $p=\infty$.
The condition $m\ge N_\beta$ leaves open what happens at low-order derivatives at zeros of $G$; testing finite-rank examples for algebraic relations involving $G^{(r)}(\beta)$ with $r<N_\beta$ would show whether the omitted derivatives are genuinely dependent or just outside the method.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper's new object is the two-variable entire function Theta(x,y), and the announced result would indeed extend Tanaka's theorems by making the values and all mixed partial derivatives at algebraic points algebraically independent. The reduction of Theorem 5 to Theorem 6 via polynomial/linear transformations over Q is the best part of the work: the triangular matrix argument is carefully done and appears sound. The construction itself is worth preserving.

The problem is in the Mahler machinery, and it is load-bearing. Lemma 6 asserts that the point gamma = (1,...,1,a) satisfies conditions (I), (II), (III)_p, and (IV)_p for the companion matrix Omega. Condition (III)_p requires log |alpha_i^{(k)}|_p <= -c rho^k for every coordinate i, with c > 0. For any coordinate equal to 1, the left-hand side is 0 and the right-hand side is negative, so the inequality is impossible. This is not a minor technicality: Proposition 3 in the proof of Theorem 9 uses (III)_p to bound |(Omega^k gamma)^xi|_p for arbitrary multi-indices xi, and that bound fails when one coordinate is 1. The application of Theorem 9 to this point therefore collapses, in both the complex and p-adic cases. The stress-test note's concern about Lemma 2 is real but understates things: Lemma 2, as stated, would imply (III)_p for points with coordinate 1, which is impossible; the lemma is not merely unproved but false in the regime the proof needs.

There are also softer gaps: Theorem 8 is stated without proof, and the proof of Theorem 7 is only sketched as a combination of Nishioka's proofs. Those are fillable, but they add to the required revision.

On balance: do not accept the paper in its current form. The construction and the reduction deserve referee attention, and a careful referee could identify exactly how the Mahler conditions must be modified or what alternative point could serve. This is a conditional reject with a clear path, not a rejection of the underlying idea.

Referee Report

3 major / 4 minor

Summary. The paper constructs, for each p equal to infinity or a prime, a two-variable entire function Theta(x,y) on C_p^2 generated by a linear recurrence {R_k}, and claims (Theorem 5) that the values and all partial derivatives of Theta at all algebraic points (alpha,beta) with alpha nonzero and beta algebraic, together with the values G^{(N_beta)}(beta), form an algebraically independent set over Q. The proof reduces Theorem 5 to Theorem 6 by a linear-recurrence shift, then proves Theorem 6 from a new multiparameter Mahler-function criterion (Theorem 7). Theorem 7 is assembled from an algebraic independence theorem for Mahler functions over rational function fields (Theorem 8) and a value-independence theorem (Theorem 9, proved in Section 3.4), and the final obstruction is ruled out by two lemmas of Tanaka (Lemmas 7 and 8) plus an elementary partial-fraction argument.

Significance. If the proof is completed, the result is a notable explicit construction: a single entire function of two variables whose values and all partial derivatives at every algebraic point (with nonzero first coordinate) are algebraically independent, in both the complex and p-adic settings. This unifies and refines the one-variable theorems of Nishioka and Tanaka, and the reduction to Mahler's method is a sensible and potentially useful architecture. The paper does not contain machine-checked proofs or reproducible code, but the main construction and the reduction steps are explicit and the reliance on Mahler-function theory is well motivated. The main barriers are two unproved foundational statements, Lemma 2 and Theorem 8, whose proofs are asserted to follow by combining or adapting earlier arguments; these gaps are load-bearing and must be addressed before the claims can be accepted.

major comments (3)

[Section 3.2, Lemma 2] Lemma 2 is the p-adic Mahler vanishing theorem on which the prime-p case of Theorem 5 rests, via Lemma 6 and Theorem 9. The manuscript does not prove this lemma; it states that it 'can be proved in the same way as in the proof of Theorem 2.2 in Nishioka [7]'. This is not a proof, and the application is not a routine transcription: the point gamma = (1,...,1,a) of (31) has coordinates equal to 1, so the verification of condition (III)_p (the p-adic decay of every coordinate of Omega^k gamma) and condition (IV)_p requires a specific argument showing that the cofactor inequality sum |A_i1| log|alpha_i|_p < 0 implies the required bounds even when some alpha_i = 1. Please provide a complete proof of Lemma 2, or a precise statement of a published theorem whose hypotheses are verified for the companion matrix (27) and the point gamma.
[Section 3.3, Theorem 8] Theorem 8 is stated as the first half of the proof of the paper's central new criterion, Theorem 7. The text says it 'can be obtained by combining the proof of Theorem 3 in Nishioka [6] and the second half of that of Theorem 3.5 in Nishioka [7]', but no proof or derivation is given. Since Theorem 7 is used in the proof of Theorem 6 for both the complex and p-adic cases, and since Theorem 8 is not a quotation of a single published theorem with matching hypotheses, this omission leaves a load-bearing gap. Please supply a full proof of Theorem 8, or state exact theorem numbers and verify that the hypotheses of those theorems match the linear-recurrence setup of Section 4, including the functional equations (32) and (33).
[Section 4, proof of Theorem 6, final paragraph] The final step says that (34) 'does not hold since beta_j are nonzero distinct numbers and since c_jm are not all zero'. This is correct but terse; the proof should spell out the partial-fraction argument: after multiplying by the common denominator, the distinct poles at X = 1/beta_j force all coefficients c_jm to vanish, and the polynomial part then forces the constant delta to be zero. I do not believe this is a substantive error, but a sentence of justification would make the contradiction fully transparent.

minor comments (4)

[Abstract/Introduction] The word 'algbraic' appears in the opening sentence of the abstract; please correct the typo.
[Section 2, notation] The shifted recurrence is written as '~R_k := R_{k+k0}' and then used with a tilde over the function symbols; the notation is understandable but the tilde is easy to lose in print. Consider using a more visible notation, e.g. R^{(k0)}_k or a subscript, to avoid confusion.
[Corollary 2] The phrase 'at any distinct algebraic points (alpha,beta) with alpha != 0' is ambiguous; the corollary actually asserts algebraic independence of the whole infinite set of derivatives at all algebraic points, and 'distinct' is not needed. Please rephrase.
[Section 3.2, Lemma 2] Since Lemma 2 is a p-adic analogue of Mahler's vanishing theorem, a precise reference to the exact theorem in Nishioka's book or article, with the statement reproduced, would help the reader verify that the cited argument indeed covers the case where some coordinates of alpha equal 1.

Circularity Check

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No significant circularity: the proof is a reduction to established external Mahler-function criteria and does not fit parameters or assume the conclusion.

full rationale

The derivation chain is linear and external: Theorem 5 is reduced to Theorem 6 through polynomial and linear changes of variables (equations (9)-(14)), and Theorem 6 is proved by applying Theorem 7, a Mahler-value criterion, together with Lemmas 7 and 8 from Tanaka [11] and Lemma 6 (complex case from Tanaka [10]; prime case via Lemma 2 from Nishioka [7]). The objects Theta, H, F_m, and G are defined independently by series and products, and no parameter is fitted to the target algebraic-independence set; the conclusion is not an input by construction. The cited results are published external theorems, not self-citations by the present author, and they do not presuppose Theorem 5. Two passages assert omitted proofs (Lemma 2 and Theorem 8, 'can be proved in the same way as...' and 'can be obtained by combining...'); these are gaps or reliance on external results, hence correctness risks, not circularity. The absence of Lemma 2's proof in the prime case is a limitation, but it does not make any equation of the paper equivalent to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted constants. It relies on the standing condition (R)_p and on several external results: Masser's vanishing theorem, Nishioka's Mahler-function theory, and Tanaka's rational-function lemmas. The most fragile is Lemma 2, a p-adic vanishing theorem whose proof is only sketched by analogy.

assumptions (4)

domain assumption The recurrence {R_k}_{k>=0} satisfies condition (R)_p: in the complex case Phi(plus or minus 1) != 0, no ratio of distinct roots is a root of unity, and the sequence is not geometric; in the p-adic case Phi is irreducible and has a dominant root.
This is the standing hypothesis of Theorems 5 and 6. It is needed for Lemma 6 (Mahler conditions) and for Tanaka's Lemmas 7/8. The theorem is not claimed outside this condition, and the geometric-progression case is known to fail for Theorem 4.
domain assumption The p-adic vanishing theorem (Lemma 2), the analogue of Mahler's vanishing theorem, is valid.
Lemma 2 is used to prove Lemma 6 in the p-adic case and hence to apply Theorem 9. The paper says it 'can be proved in the same way as in the proof of Theorem 2.2 in Nishioka [7]' but does not reproduce the proof.
domain assumption Tanaka's Lemmas 7 and 8 (special cases of Theorems 1 and 2 of Tanaka [11]) are correct.
These lemmas are quoted without proof; they are the decisive tool that rules out the functional-equation cases (i) and (ii) in the proof of Theorem 6.
standard math The cited results of Nishioka [6,7], Masser [4], and Tanaka [9,10] on Mahler functions and algebraic independence are accepted as correct.
The proof relies on these external theorems without reproducing them, including Lemma 3 and the theory of algebraic independence of Mahler functions.

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Pith. "Pith review of Algebraic independence of certain entire functions of two variables generated by linear recurrences." pith.science (2026). https://pith.science/paper/Z6ICEXGW

@misc{pith2026190806289,
  author       = {Pith},
  title        = {Pith review of: Algebraic independence of certain entire functions of two variables generated by linear recurrences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z6ICEXGW}},
  note         = {Machine review of arXiv:1908.06289}
}

read the original abstract

In this paper we construct an entire function of two variables having the property that its values and its partial derivatives of any order at any distinct algebraic points are algebraically independent. Such an entire function is generated by a linear recurrence. In order to prove this result, we reduce the algebraic independency to that of Mahler functions of several variables by shifting the linear recurrence and apply the theory of Mahler functions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[7]

Nishioka, Mahler Functions and Transcendence, Lecture Not es in Mathe- matics No

K. Nishioka, Mahler Functions and Transcendence, Lecture Not es in Mathe- matics No. 1631, Springer, 1996

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[6]

Nishioka, Algebraic independence of Mahler functions and their value s, To- hoku Math

K. Nishioka, Algebraic independence of Mahler functions and their value s, To- hoku Math. J. 48 (1996), 51–70

work page 1996
[1]

F. R. Gantmacher, Applications of the Theory of Matrices, New Y ork, Inter- science, 1959

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[2]

K. K. Kubota, On the algebraic independence of holomorphic solutions of c er- tain functional equations and their values , Math. Ann. 227 (1977), 9–50

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[3]

Mahler, Arithmetische Eigenschaften der L¨ osungen einer Klasse vo n Funk- tionalgleichungen, Math

K. Mahler, Arithmetische Eigenschaften der L¨ osungen einer Klasse vo n Funk- tionalgleichungen, Math. Ann. 101 (1929), 342–366

work page 1929
[4]

D. W. Masser, A vanishing theorem for power series , Invent. math. 67 (1982), 275–296

work page 1982
[5]

Nishioka, Algebraic independence of certain power series of algebrai c num- bers, J

K. Nishioka, Algebraic independence of certain power series of algebrai c num- bers, J. Number Theory 23 (1986), 353–364

work page 1986
[8]

A. B. Shidlovskii, Transcendental Numbers, Walter de Gruyter, New York, 1989

work page 1989

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[9]

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Waldschmidt, Diophantine Approximation on Linear Algebraic Gr oups, Grundlehren der mathematischen Wissenschaften No

M. Waldschmidt, Diophantine Approximation on Linear Algebraic Gr oups, Grundlehren der mathematischen Wissenschaften No. 326, Springer, 2000. Department of Mathematics, Faculty of Science and Technology, K eio University 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan E-m...

2000

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Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [7]

Nishioka, Mahler Functions and Transcendence, Lecture Not es in Mathe- matics No

K. Nishioka, Mahler Functions and Transcendence, Lecture Not es in Mathe- matics No. 1631, Springer, 1996

work page 1996

[2] [6]

Nishioka, Algebraic independence of Mahler functions and their value s, To- hoku Math

K. Nishioka, Algebraic independence of Mahler functions and their value s, To- hoku Math. J. 48 (1996), 51–70

work page 1996

[3] [1]

F. R. Gantmacher, Applications of the Theory of Matrices, New Y ork, Inter- science, 1959

work page 1959

[4] [2]

K. K. Kubota, On the algebraic independence of holomorphic solutions of c er- tain functional equations and their values , Math. Ann. 227 (1977), 9–50

work page 1977

[5] [3]

Mahler, Arithmetische Eigenschaften der L¨ osungen einer Klasse vo n Funk- tionalgleichungen, Math

K. Mahler, Arithmetische Eigenschaften der L¨ osungen einer Klasse vo n Funk- tionalgleichungen, Math. Ann. 101 (1929), 342–366

work page 1929

[6] [4]

D. W. Masser, A vanishing theorem for power series , Invent. math. 67 (1982), 275–296

work page 1982

[7] [5]

Nishioka, Algebraic independence of certain power series of algebrai c num- bers, J

K. Nishioka, Algebraic independence of certain power series of algebrai c num- bers, J. Number Theory 23 (1986), 353–364

work page 1986

[8] [8]

A. B. Shidlovskii, Transcendental Numbers, Walter de Gruyter, New York, 1989

work page 1989