Pith. sign in

REVIEW 4 major objections 5 minor 15 references

A simultaneous approximation problem for exponentials and logarithms

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Two coprime integer polynomials cannot both be extremely small at a log-exponential triple.

desk verdict The triple extension of Brownawell is new and the architecture is plausible, but two load-bearing proof gaps (undefined b0 and an unproved substitution-invariance) make the derivation unverifiable as written. read the letter →

arxiv 2505.20957 v1 pith:RWDIQKJX submitted 2025-05-27 math.NT

Veekesh Kumar , Riccardo Tosi This is my paper
classification math.NT MSC 11J8211J85
keywords simultaneousapproximationalgebraicindependencemeasuretranscendenceexponentialandlogarithmicvaluesauxiliaryfunctionssemi-resultantsSiegel'slemma
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 aims to prove that the three numbers $\log\alpha_2/\log\alpha_1$, $\alpha_1^\beta$, and $\alpha_2^\beta$—where $\alpha_1,\alpha_2$ are multiplicatively independent algebraic numbers and $\beta$ is a quadratic irrational—cannot be approximated too well simultaneously by values of two coprime integer polynomials. Its main theorem shows that if coprime $P(x,y)$ and $Q(x,y,z)$ make $\log\max(|P(\log\alpha_2/\log\alpha_1,\alpha_1^\beta)|,|Q(\log\alpha_2/\log\alpha_1,\alpha_1^\beta,\alpha_2^\beta)|)\le -r^C$ for the size parameter $r$, then a nonzero integer polynomial $U(x)$ must exist with an unusually small value at $\log\alpha_2/\log\alpha_1$ and controlled degree plus log-height. From that, the paper derives an explicit lower bound for the simultaneous values and a quantitative algebraic-independence measure for any two of the three numbers. A sympathetic reader would care because the result turns an algebraic-independence fact into a fully quantitative diophantine statement with a proof method that can be checked step by step.

What carries the argument

The load-bearing object is the auxiliary function $F(z)=\sum_{\lambda}\psi(\lambda)z^{\lambda_1}\alpha_1^{\lambda_2 z}\alpha_1^{\beta\lambda_3 z}$, whose coefficients are chosen by an integer linear-system lemma so that $F$ vanishes at the grid points $z_\mu=\mu_0+\mu_1 d^2\beta+\mu_2(\log\alpha_2/\log\alpha_1)+\mu_3(\log\alpha_2/\log\alpha_1)d^2\beta$. The proof then runs on a dichotomy: either some $F(z_\mu)$ is not too small, which starts a chain of semi-resultants (resultant-like quantities that eliminate one variable while keeping degree and height bounds) peeling off the variables $z$ and $y$ to produce a small univariate polynomial in $\log\alpha_2/\log\alpha_1$; or every coefficient $\psi(\lambda)$ is tiny, forcing a modified function $G$ and a second elimination round. Semi-resultants are the quantitative engine: they bound degrees and heights while transferring smallness at a point. The final rigidity argument compares the resultants of consecutive irreducible factors $t_N$ and $t_{N+1}$, showing the same factor $u$ works for all $N$ and can be evaluated at two different scales of $N$.

What would settle it

Take small explicit data, for instance $\alpha_1=2$, $\alpha_2=3$, $\beta=\sqrt{2}$ with small degrees for $P$ and $Q$, solve the linear system that determines the coefficients $\Psi_\lambda$ at the numerical point, and then check symbolically whether the resulting expression $S_\mu(x,y,z)$ is the zero polynomial in $x,y,z$. If any $S_\mu$ is nonzero, the Case 2 inference $S_\mu(\xi_1)=0$ fails; if the identity holds for several parameter choices, that supports the step.

Watch

Extended reading notes

Core claim

The central claim, Theorem 1.1, is that for multiplicatively independent algebraic $\alpha_1,\alpha_2$ and quadratic irrational $\beta$, if coprime $P(x,y),Q(x,y,z)\in\mathbb{Z}[x,y,z]$ satisfy $\log\max(|P(\log\alpha_2/\log\alpha_1,\alpha_1^\beta)|,|Q(\log\alpha_2/\log\alpha_1,\alpha_1^\beta,\alpha_2^\beta)|)\le -r^C$, then there is a nonzero $U(x)\in\mathbb{Z}[x]$ with $\log|U(\log\alpha_2/\log\alpha_1)|<-r^{C/4}$ and $\deg U+\log H(U)<r^{C/C_1}$. The proof constructs an auxiliary function with many prescribed zeros, uses a coefficient-versus-value dichotomy to pass to an elimination chain, and peels off the variables one by one with semi-resultants until only a univariate polynomial in $\log\alpha_2/\log\alpha_1$ remains. Theorem 1.2 follows by contradiction with known one-variable lower bounds: no coprime pair can actually attain the tiny values assumed, so $\log\max(|P|,|Q|)>-r^{C_2}$. Corollary 1.1 then gives a measure of algebraic independence for any two of $\log\alpha_2/\log\alpha_1$, $\alpha_1^\beta$, and $\alpha_2^\beta$.

Load-bearing premise

The proof assumes that replacing the exponential $\alpha_1^\beta$ by the nearby algebraic root $\xi_1$ preserves the vanishing of the auxiliary expressions, even though the equations were solved only at the original numerical point.

Editorial extensions

If this is right

  • For every coprime pair $P(x,y)$, $Q(x,y,z)$, the simultaneous values satisfy $\log\max(|P(\log\alpha_2/\log\alpha_1,\alpha_1^\beta)|,|Q(\log\alpha_2/\log\alpha_1,\alpha_1^\beta,\alpha_2^\beta)|)>-r^{C_2}$, so super-small simultaneous approximation is impossible.
  • Any two of the three numbers are algebraically independent, with the quantitative bound $\log|P(\theta_1,\theta_2)|>\exp(-C\deg_y P(\deg_y P+\deg_x P)^2(2\deg_y P+\deg_x P+\log H(P)))$.
  • The same proof yields a univariate polynomial that is small at $\alpha_1^\beta$ instead of at the logarithm ratio, so the lower-bound argument can run with any one of the three coordinates.
  • The resultant-based remarks extend the lower bound to coprime polynomial pairs in which both polynomials involve the same variable, as long as that variable is eliminated first.

Reading between the lines

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

  • A direct symbolic check of the Case 2 substitution step—whether the linear-system solution remains a formal identity after replacing $\alpha_1^\beta$ by $\xi_1$—would give a clean verification of that load-bearing assertion.
  • If the substitution step holds, the same auxiliary-function-and-elimination architecture should adapt to triples of the shape $\log\alpha_2/\log\alpha_1$, $\alpha_1^\beta$, $\alpha_2^\beta$ with $\beta$ of higher algebraic degree, provided the zero-grid and coefficient estimates remain valid.
  • The consecutive-resultant rigidity mechanism—showing that a degree-and-height controlled small value at one scale forces an actual irreducible factor valid at all scales—could be reused in other elimination problems.
  • The constants in the bounds are effectively computable but astronomically large, so the practical value of the theorem is structural: it identifies a univariate obstruction that any simultaneous approximation must produce.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper claims a quantitative simultaneous approximation theorem (Theorem 1.1) for the three numbers log α2/log α1, α1^β and α2^β, where α1, α2 are multiplicatively independent algebraic numbers and β is a quadratic irrational. It states that if two coprime integer polynomials P(x,y) and Q(x,y,z) are simultaneously extremely small at that point, then there exists a nonzero integer polynomial U(x) whose value at log α2/log α1 is very small, with degree and height controlled in terms of the parameters of P and Q. From this it derives a lower bound for simultaneous approximation (Theorem 1.2) and a measure of algebraic independence (Corollary 1.1). The proof follows Brownawell's method: an auxiliary exponential polynomial F is constructed by Siegel's lemma, Schwarz's lemma and Tijdeman's lemma are used to force a dichotomy, and semi-resultants are used to eliminate variables. The manuscript is a genuine derivation from cited lemmas; there is no circularity and no fitted parameters, but several load-bearing steps are asserted without proof.

Significance. If correct, the result would be a substantial and natural extension of the Gelfond-Feldman and Brownawell measures to the setting of log α2/log α1 together with α1^β and α2^β, and it would yield the stated algebraic independence measure. The paper has the right general architecture and makes the parameter bookkeeping explicit. Its main value, however, is contingent on the proof being valid. The manuscript does not provide machine-checked proofs or reproducible code, and the current version contains gaps in both cases of the central dichotomy, so the theorems are not established as written.

major comments (4)
  1. [Lemma 2.4; §4 Step 1] The quantity b0 appears in the statement of Lemma 2.4 in the factor (72c2kR/(b0(kR+1)^2))^{(kR+1)^4}, but it is never defined in the lemma. In §4 Step 1 the proof says 'we bound a0, b0 ≥ 1' and uses this to discard the factor. This is problematic in two ways. First, a0 is defined in Lemma 2.4 as min(1, |γ−γ'|), so a0 ≤ 1 by definition; asserting a0 ≥ 1 requires all nonzero differences |γ_{λ2,λ3} − γ_{λ2',λ3'}| to be at least 1, which is not shown and is generally false for large D2. Second, if b0 is the minimal pairwise distance among the interpolation points zµ, which is the natural reading since the zµ are renamed β0,...,β_{s−1}, then for zµ = μ0+μ1d²β+μ2 log α2/log α1 + μ3 (log α2/log α1)d²β with 0 ≤ μi ≤ kN, there are (kN+1)^4 points in an interval of length O(N); by the pigeonhole principle the minimal spacing is O(N^{-3}) < 1 for large N. Hence the step '72c2kN/(b0(kN+1)^2) < 1' has no basis, and the dichotomy at the start of §4, on which both Case 1 and Case 2 rely, is unsupported.
  2. [§4, Case 2] The key claim that 'the expression obtained from F(zµ) by replacing every occurrence of α1^β with ξ1 still vanishes' is not justified. The coefficients Ψλ were obtained as a solution of the linear system F(zµ)=0 evaluated at the specific point (log α2/log α1, α1^β, α2^β); the equations are not polynomial identities in the formal variables x,y,z. The root ξ1 is chosen as the closest root of P(log α2/log α1, y) to α1^β, not as a Galois conjugate of α1^β, so substituting α1^β by ξ1 is not a field homomorphism of the ring in which the system was solved. Without Sµ(ξ1)=0, the estimate log |Sµ(α1^β)| ≤ −(1/2)r^{7C/8} does not follow, and the whole Case 2 construction of the polynomial that replaces A3 collapses.
  3. [§4, Case 1] In the passage defining A2 as the norm of A1 over Q(log α2/log α1, ξ1, ξ2), the proof asserts that for every automorphism σ_i of the Galois closure over that base field, σ_i(A1) can be controlled because 'the coefficients ψ(λ) of F are invariant under σ_i' and hence σ_i(F(zµ)) = F(σ_i(zµ)). But σ_i need only fix log α2/log α1, ξ1 and ξ2; the coefficients ψ(λ) lie in Z[log α2/log α1, α1^β, α2^β] and are not invariant under such σ_i. Moreover, A1 was formed after replacing α1^β and α2^β by ξ1 and ξ2, so σ_i(A1) is not obtained by applying σ_i to the original expression F(zµ). The conclusion that all conjugates have the same small modulus, and therefore log |A2| ≤ −(ε/5)δN^4 log N, is unsupported. This smallness is needed to obtain the polynomial A4 in Case 1.
  4. [§4, Case 2] The sentence 'The fact that the Ψλ's do not have a common factor implies that the polynomials Ψλ(log α2/log α1, ξ1, z) are not all zero' introduces an unproved fact. The Ψλ are components of one non-trivial solution of a linear system produced by Siegel's lemma; nothing in the construction guarantees that the tuple (Ψλ) is coprime as polynomials, and a common factor that vanishes at the special point would not be removable. Since the Case 2 argument needs some Ψλ(log α2/log α1, ξ1, z) to be nonzero, this missing justification is load-bearing.
minor comments (5)
  1. [§3, Step 1] The condition '0 ≤ μi ≤ N for 0 ≤ i ≤ 4' should be '0 ≤ i ≤ 3'; the tuple μ has only four coordinates.
  2. [§3, Step 1] In the displayed formula for aλ,μ, the expression 'log α2/log α d2β' is missing a subscript; it should be log α1.
  3. [Lemma 2.4] The notation β0,...,β_{s−1} for the interpolation points collides with the quadratic irrational β used throughout the paper; these points should be renamed.
  4. [§4, Step 2] The resultant is first written as r_N of t_{N−1} and t_N, but the next sentence refers to t_N and t_{N+1}; the indexing should be made consistent.
  5. [§4, Case 2] In the Hermite interpolation estimate, the displayed lower bound |t−zµ| ≥ N^2 − N should be N^2 − O(N log N) if the range |μi| ≤ N log N is used; the exponential conclusion is unchanged, but the range of μ should be stated consistently.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof is a standard auxiliary-function derivation from external lemmas, with no fitted inputs or self-citation chains.

full rationale

The derivation of Theorem 1.1 is self-contained in the sense relevant to circularity. The auxiliary function F is constructed by Siegel's lemma from a linear system whose coefficients depend on the same transcendental numbers that occur in P and Q; this is the standard auxiliary-function method, not a renaming of the target conclusion. The estimates for F come from Schwarz's lemma and from Tijdeman's Lemma 2.4, an external result, and the subsequent elimination steps use resultants/semi-resultants and Lemma 2.7. No parameter is fitted to the small values of P and Q, and the final polynomial U is not an input but is produced by the resultant process. The proof of Theorem 1.2 combines Theorem 1.1 with Waldschmidt's Theorem 5.1 by contradiction; this is a normal derivation, not a prediction that reduces to its inputs. The bibliography contains no self-citations by Kumar and Tosi, so no load-bearing self-citation chain is present. Potential issues noted in the manuscript, such as the undefined b0 in Lemma 2.4 and the substitution-invariance assertion in Case 2 of Section 4, are correctness or rigor concerns, not circularity: even if those steps fail, the failure would be a false lemma application or an unproved polynomial identity, not the target theorem being assumed as an input.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The proof rests on standard lemmas from transcendence theory. No free parameters are fitted to data, and the paper introduces no new particles, forces, dimensions, or other postulated entities. The constants C, C1, ε, N, D1, D2 are chosen existentially in the proof, not fitted.

assumptions (8)
  • standard math Gelfond-Schneider theorem: α^β is transcendental for algebraic α≠0,1 and algebraic irrational β.
    Used in Section 3, Step 1 to assert that log α2/log α1, α1^β, α2^β are transcendental.
  • standard math Siegel's lemma (Lemma 2.1, from Brownawell [2]) provides small-height solutions to linear systems.
    Used in Section 3 to construct the auxiliary function F.
  • standard math Schwarz lemma (Lemma 2.3) controls a holomorphic function with many zeros.
    Used in Section 3, Step 2 to derive smallness of F on a large ball.
  • standard math Tijdeman's auxiliary result (Lemma 2.4, from [8]) is quoted as stated.
    Used in Section 4, Step 1 to obtain a dichotomy on the coefficients of F. The statement as printed contains an undefined parameter b0.
  • standard math Semi-resultant lemmas (Lemmas 2.5 and 2.6, from Brownawell [3]) bound degrees, heights, and values of resultants.
    Used in the elimination steps producing A3 and A4.
  • standard math Factor lemma (Lemma 2.7, from Brownawell [1]) extracts an irreducible factor with controlled size.
    Used in Section 4, Step 2 to pass from sN to tN.
  • standard math Waldschmidt's transcendence measure (Theorem 5.1, from [14]) gives lower bounds for univariate integer polynomials at α^β and at log α2/log α1.
    Used in the proof of Theorem 1.2 to contradict the small value produced by Theorem 1.1.
  • domain assumption Multiplicative independence of α1, α2 and quadratic irrationality of β are the domain assumptions of the theorems.
    These hypotheses are stated in Theorem 1.1 and used throughout the construction.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A simultaneous approximation problem for exponentials and logarithms." pith.science (2026). https://pith.science/paper/RWDIQKJX

@misc{pith2026250520957,
  author       = {Pith},
  title        = {Pith review of: A simultaneous approximation problem for exponentials and logarithms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RWDIQKJX}},
  note         = {Machine review of arXiv:2505.20957}
}
abstract

Let $\alpha_1,\alpha_2$ be non-zero algebraic numbers such that $\frac{\log \alpha_2}{\log\alpha_1}\notin\mathbb{Q}$ and let $\beta$ be a quadratic irrational number. In this article, we prove that the values of two relatively prime polynomials $P(x,y,z)$ and $Q(x,y,z)$ with integer coefficients are not too small at the point $\left(\frac{\log\alpha_2}{\log \alpha_1},\alpha_1^\beta, \alpha_2^\beta \right)$. We also establish a measure of algebraic independence of those numbers among $\frac{\log\alpha_2}{\log \alpha_1}$, $\alpha^\beta_1$ and $\alpha^\beta_2$ which are algebraically independent.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

  1. [1]

    W. D. Brownawell, On the Gelfond-Feldman measure of algebraic independence, Compositio Math., 38 (1979), no. 3, 355-368

    work page 1979

  2. [2]

    W. D. Brownawell, Gelfond’s Method for Algebraic Independence, Trans. Amer. Math. Soc., 210 (1975), 1-26

    work page 1975

  3. [3]

    W. D. Brownawell, Some remarks on semi-resultants, Advances in Transcendence Theory, Academic Press (1977), 205-210

    work page 1977

  4. [4]

    W. D. Brownawell and M. Waldschmidt, The algebraic independence of certain numbers to algebraic powers, Acta Arith., 32 (1977), 63-71

    work page 1977

  5. [5]

    A. O. Gelfond, Transcendental and algebraic numbers, GITTL, Moscow,1952 [English translation: Dover, New York, 1960]

    work page 1952

  6. [6]

    A. O. Gelfond and N. I. Feldman, On the measure of relative transcendence of certain numbers, Izv. Akad. Nauk SSSR, Ser. mat., 14 (1950), 493-500

    work page 1950

  7. [7]

    Lang, Complex Analysis, GTM 103, Springer-Verlag, 1999

    S. Lang, Complex Analysis, GTM 103, Springer-Verlag, 1999

    work page 1999

  8. [8]

    Tijdeman, An auxiliary result in the theory of transcendental numbers, J

    R. Tijdeman, An auxiliary result in the theory of transcendental numbers, J. Number Theory, 5 (1973), 80-94

    work page 1973

Show all 15 references
  1. [9]

    Tijdeman, On the algebraic independence of certain numbers, Indag

    R. Tijdeman, On the algebraic independence of certain numbers, Indag. Math., 33 (1971), 146-162

    1971

  2. [10]

    Yu. V. Nesterenko, Algebraic Independence(2011) (New Delhi: Narosa Publishing House)

    2011

  3. [11]

    N. I. Feldman and A. B. Sidlovskii, The development and present state of the theory of transcendental numbers, English transl., Russian Math. Surveys, 22 (1967),1-79

    1967

  4. [12]

    A. A. Shmelev, The algebraic independence of certain transcendental numbers, Mat. Zametki, 3 (1968), 51-58

    1968

  5. [13]

    Tubbs, A diophantine problem on elliptic curves, Trans

    R. Tubbs, A diophantine problem on elliptic curves, Trans. Amer. Math. Soc., 309 (1988), no. 1, 325-338

    1988

  6. [14]

    Waldschmidt, Transcendence measures for exponentials and logarithms, J

    M. Waldschmidt, Transcendence measures for exponentials and logarithms, J. Austral. Math. Soc. (Series A) 25 (1978), 445-465. 20 VEEKESH KUMAR AND RICCARDO TOSI

    1978

  7. [15]

    Waldschmidt, Diophantine approximation on linear algebraic groups, Grundlehren der mathema- tischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol

    M. Waldschmidt, Diophantine approximation on linear algebraic groups, Grundlehren der mathema- tischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 326, Springer-Verlag, Berlin, 2000. (Veekesh Kumar) Department of Mathematics, Indian Institute of Tech...

    2000

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.