Comments on J. F. Ritt's book "Integration in Finite Terms"

Askold Khovanskii

REVIEW 4 major objections 6 minor 13 references

Comments on J. F. Ritt's book "Integration in Finite Terms"

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper modernizes Liouville's theorems and proves that an n-th order homogeneous linear differential equation is solvable by generalized quadratures exactly when it possesses an algebraic-exponential solution and the reduced equation…

desk verdict A useful modern companion to Ritt with a genuinely new-looking Puiseux proof of Rosenlicht's criterion, but that proof is conditional on an unproved generalized Newton theorem and the internal references need cleanup. read the letter →

arxiv 1908.02048 v1 pith:JHDJTGPV submitted 2019-08-06 math.AG

Askold Khovanskii This is my paper · ORCID

classification math.AG MSC 12H0534A3034M1514H0532S40

keywords Liouville'stheoremsgeneralizedquadraturesdifferentialfieldsPuiseuxseriestopologicalGaloistheorymonodromysolvabilityinfinitetermslinearequations

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 gives modern proofs of the two classical Liouville theorems and extends the second one to homogeneous linear differential equations of arbitrary order. The central result is an if-and-only-if criterion: an n-th order equation over a functional differential field is solvable by generalized quadratures exactly when it has a solution of the form $\exp z$ with $z'$ algebraic over the coefficient field and the reduced $(n-1)$-st order equation is then solvable over the enlarged field. The proof is elementary in the sense that it avoids valuation theory and instead uses convergent Puiseux series expansions furnished by a generalized Newton theorem. The same criterion is shown to hold for nonlinear homogeneous equations satisfying a weighted-degree condition. The second half of the paper outlines a topological Galois theory in which branching and monodromy, rather than differential Galois groups, obstruct solvability in finite terms.

What carries the argument

The mechanism that carries the proof is the generalized Newton theorem, stated as Theorem 2.15: an algebraic function over a field of meromorphic functions on a Riemann surface has convergent Puiseux expansions at infinity with coefficients in a single finite extension of that field. From it, the paper builds a one-parameter family of solutions in a generalized extension by an integral or an exponential of an integral, expands the family in Puiseux series in the varying transcendental element, and plugs the expansion into the differential equation. The leading-term analysis then forces the leading coefficient of the expansion to satisfy the original equation, which is precisely the algebraic solution required by the criterion. This machinery lets the argument avoid valuation theory and remain inside functional differential fields of meromorphic functions.

What would settle it

Exhibit an algebraic function defined by a polynomial over a functional differential field $K$ whose Puiseux expansion at infinity requires coefficients that generate an infinite extension of $K$, or exhibit a homogeneous linear equation over $K$ that is solvable by generalized quadratures but violates condition (1) of Theorem 2.12 by having no solution $\exp z$ with $z'$ algebraic over $K$.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that Liouville's Second Theorem is not a second-order fact. For any order $n$, a homogeneous linear differential equation $y^{(n)}+a_1y^{(n-1)}+\cdots+a_ny=0$ with coefficients in a functional differential field $K$ is solvable by generalized quadratures over $K$ if and only if it has a solution $y_1=\exp z$ with $z'$ algebraic over $K$ and the reduced equation of order $n-1$ over $K(y_1)$ is solvable by generalized quadratures over $K(y_1)$. The proof replaces the standard Picard-Vessiot and valuation-theoretic route with a Puiseux-series argument: a solution living in an extension by an integral or by an exponential of an integral generates a one-parameter family of solutions, and the leading term of the family's Puiseux expansion must already supply an algebraic solution of the associated generalized Riccati equation. The same leading-term argument, together with a weighted-degree condition, extends the criterion to nonlinear homogeneous equations. For second-order equations the statement reduces to the classical form of Liouville's Second Theorem.

Load-bearing premise

The load-bearing premise is the generalized Newton theorem, stated without proof in Section 2.4.3: an algebraic function over a field of meromorphic functions has convergent Puiseux expansions at infinity whose coefficients lie in one finite extension of that field, uniformly over the punctured Riemann surface.

Editorial extensions

If this is right

For any $n$, solvability by generalized quadratures forces an exponential solution $\exp z$ whose logarithmic derivative $z'$ is algebraic over the coefficient field; transcendental exponential solutions can only appear after an algebraic adjunction.
Solvability therefore becomes a recursive finite-step check: find the algebraic exponential solution, reduce order, and repeat on the new coefficient field.
For $n=2$ the criterion automatically returns Liouville's Second Theorem, since first-order equations are solvable by quadratures.
Nonlinear homogeneous equations with a weighted-degree-dominant monomial obey the same dichotomy, so the criterion is not limited to linear equations.
The paper's proof makes the theorem available over functional fields of meromorphic functions without a side trip through abstract differential algebra and valuation theory.

Reading between the lines

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

Although the paper does not spell this out, the recursive form of the criterion looks like a decision procedure: for each candidate algebraic extension, the Puiseux leading-term computation could be turned into an algorithm that either produces the required solution or terminates with a certificate of non-solvability.
The same family-of-solutions and leading-term argument might extend to equations with irregular singular behavior, where the uniform Puiseux expansion of the generalized Newton theorem does not apply and no analogous elementary proof is currently available.
A testable consequence of the paper's viewpoint is that any solution representable by generalized quadratures but not by algebraic exponentials would force the equation to become solvable after adjoining such an algebraic exponential; this can be checked on explicit transcendental equations whose solutions have no obvious algebraic-exponential entry point.
The topological Galois theory outline suggests that, for Fuchsian equations, the only obstructions to solvability in finite terms are branching obstructions; completing the unpublished proofs in the later sections would give a unified 'only branching matters' explanation for many non-solvability results.

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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

Dear colleague,

What you should know: this is Khovanskii's comments on Ritt's book, but you can read it as two working papers. Section 2 contains the substance. Section 2.3 gives a modern proof of Liouville's First Theorem by Lie group actions and invariant 1-forms. Section 2.4 gives a new-looking proof of Rosenlicht's criterion (Theorem 2.12) that avoids valuation theory, replacing it with Puiseux expansions of a one-parameter family of solutions, plus a nonlinear generalization (Theorem 2.14). Section 3 is a survey of topological Galois theory, with most proofs deferred to his book [Kho14].

The new material is genuinely useful. The Lie-group proof of the First Theorem is clear and really does illuminate the old argument. The Puiseux route to Rosenlicht is a real change of method, and Theorem 2.14 is a modest but real extension. The paper is also honest about its gaps, which counts for something.

The soft spots, in proportion. The main one is Theorem 2.15, the generalized Newton theorem. It is stated without proof, and the author says a modern proof will appear in a separate paper. That theorem is load-bearing: the Puiseux expansions (22) and (24), and therefore the leading-coefficient arguments in Theorems 2.18 and 2.19, rest on it. If the exact statement—coefficients regular on a common covering, with a continuous radius function—doesn't hold, the proof of Theorem 2.12 collapses. The result is plausible and should be standard, but the paper doesn't provide a proof or a precise reference. So the central new proof is conditional. That's exactly the author's own description, but it is still a serious incompleteness for a paper that advertises a proof of Rosenlicht's theorem.

Second, the cross-references are a mess. The text cites Lemma 22, Lemma 27, Theorem 19, Theorem 31, and others that don't match the numbering. That doesn't hurt the mathematics, but it makes refereeing harder and suggests the manuscript hasn't been cleaned up.

Third, Section 3 is not really new. It is an outline of [Kho14], and most theorems are stated with "see [Kho14]." That's fine as a service to the reader, but it shouldn't be judged as an independent contribution.

The citation pattern is self-heavy, but mostly legitimate: the surveyed results are genuinely from his own book and papers.

Who is this for? Someone who wants a modern map of Ritt's book, or wants to see the Rosenlicht criterion approached with power series instead of valuations. It would be a good reading-group starting point.

My recommendation: send it to peer review, but ask the author to include a proof of Theorem 2.15 or cite a precise published source, and to fix the internal references. The classical theorems are true and the new proof route is plausible; the paper should not be desk-rejected. But as it stands, the advertised contribution is conditional, so acceptance should wait for the missing theorem.

Referee Report

4 major / 6 minor

Summary. The manuscript is an extended commentary on Ritt's book. Part 1 (Section 2) gives a modern proof of Liouville's First Theorem using Lie group actions and locally invariant 1-forms, and develops a generalization of Liouville's Second Theorem: Theorem 2.12 states that a homogeneous linear ODE (12) is solvable by generalized quadratures over K exactly when it has a solution y1 = exp z with z' algebraic over K and the reduced equation is solvable over K(y1). The proof follows Rosenlicht's theorem (Theorem 2.13) and avoids valuation theory by using one-parameter families of solutions and Puiseux expansions based on a generalized Newton theorem (Theorem 2.15). Part 2 (Section 3) outlines topological Galois theory: Section 3.2 is self-contained on representability by radicals and the 13th Hilbert problem, while Sections 3.3 and 3.4 state results on S-functions, monodromy pairs, Fuchsian equations, SC-germs, and holonomic systems essentially without proofs. The paper explicitly announces that most of Section 3 is an outline and that Theorem 2.15 is deferred to a separate paper.

Significance. If Theorem 2.15 is supplied, the proof of Theorem 2.12 gives a genuinely elementary, valuation-free route to Rosenlicht's theorem and to the Liouville-Picard-Vessiot criterion for homogeneous linear equations of arbitrary order. The Lie-group proof of the First Liouville Theorem is a conceptually attractive modernization of the classical arguments. The paper is honest about its limitations: the abstract and Section 2.4.1 explicitly say that Theorem 2.15 is deferred and that most of Section 3 is statement-only. The topological Galois theory portion is a useful survey and includes a complete proof in Section 3.2 of the topological criterion for representability by radicals; the repeated references to [Kho14] are appropriate for a survey. These strengths do not remove the need to make the main new proof self-contained or explicitly conditional on an established theorem.

major comments (4)

[§2.4.3, Theorem 2.15] Theorem 2.15 is the sole justification for the Puiseux expansions (22) and (24) used in Sections 2.4.5 through 2.4.7. The theorem is stated without proof, and the text says its modern proof will appear in a separate paper. The leading-term arguments in Theorems 2.18 and 2.19, which force the exponent k/p to be zero and identify the constant term as a solution, depend on those expansions holding with coefficients regular on a common covering UP and with a continuous positive radius function r(a). As written, the proof of Theorem 2.12 is therefore conditional on an unproved statement. Please include a proof of Theorem 2.15, or give a precise published reference containing exactly this parameter-dependent statement, or state explicitly that Theorem 2.12 is conditional on it.
[§2.3.4.3, proof of Theorem 2.10] The proof of the First Liouville Theorem invokes 'Lemma 27', 'Lemma 22', 'Theorem 31', and 'Theorem 21', none of which appears in the manuscript; Section 2.3.4.1 similarly refers to 'Lemma 8' and 'Theorem 9' instead of Lemma 2.5 and Theorem 2.7. Because this proof is one of the paper's two advertised new proofs, the broken cross-references make the argument unverifiable as written. Please renumber all internal references and check every such citation.
[§2.4.2 and §2.4.7] The logical relation between Theorems 2.11, 2.12, and 2.13 is mislabeled. The generalized Riccati equation (15) satisfies the hypothesis of Rosenlicht's theorem (Theorem 2.13), not of the criterion Theorem 2.12; the sentence at the end of Section 2.4.2 saying that 'The generalized Riccati equation (15) satisfies the conditions of Theorem 2.12' should refer to Theorem 2.13. Similarly, the proof in Section 2.4.7 is titled 'Proof of Rosenlicht's theorem' and proves exactly the statement of Theorem 2.13 for equation (13), but the text calls it 'Proof of Theorem 2.12' and even 'Proof of Theorem 35'. Correct the numbering and spell out the implication chain: Rosenlicht's theorem for (15) gives Corollary 2.3, which yields Theorem 2.11; Theorem 2.12 then follows from Lemma 2.8 by induction on the order.
[§2.4.6 and §2.4.7, proof of Theorem 2.19] In the proof of Theorem 2.19 the citation 'Lemma 2.20' should be 'Lemma 2.21' in both places: the exponential-integral analogue of Lemma 2.18 is Lemma 2.21, not Lemma 2.20. As printed, the reader cannot tell which statement about leading terms is being used for the case k/p = 0. This is a local error, but it occurs in the proof of a central claim and should be corrected.

minor comments (6)

[§2.3.3.2, equation (2)] Equation (2) has a stray label 'abeleqn2 (3)' that should be removed.
[§2.3.3.2, proof of Abel's Theorem] The reference 'Theorem 19' should be Theorem 2.5, and in equation (5) the denominator 'Mj(z1)' should be 'Mi(z1)'.
[§2.2.6, Theorem 2.4] Theorem 2.4 says it 'follows from Lemma 8', but the intended reference is Lemma 2.3.
[§2.4.6, Lemma 2.21] The statement of Lemma 2.21 has minor typos: 'Piueux' should be 'Puiseux', and the condition for the leading term should be written uniformly with the notation of Lemma 2.18.
[§3] There is a global renumbering problem in Section 3: the text refers to 'Theorem 10', 'Theorem 16', 'Lemma 32', 'Corollary 38', and 'Theorem 40' where the corresponding numbered statements do not appear under those numbers. A systematic pass to fix all cross-references is needed.
[§2.2.2.2, Definition 2.1] The chain in Definition 2.1 is written 'K = F0 ⊂ ... ⊆ Fn ⊃ F'; this should presumably be 'F ⊂ Fn' to match the intended definition of an extension.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity. The new proof is conditional on an unproved but not circular generalized Newton theorem, and Section 3 is openly an outline relying on the author's prior book.

full rationale

The paper's central new derivation—the Puiseux-series proof of Rosenlicht's theorem and hence of the generalized Liouville Second Theorem—does not reduce to its own inputs. The argument in Sections 2.4.5–2.4.7 uses the generalized Riccati equation, one-parameter families of solutions in integral and exponential-integral extensions, and Puiseux expansions supplied by the stated generalized Newton theorem (Theorem 2.15). The leading-coefficient argument in Theorems 2.18 and 2.19 is a genuine mathematical deduction, not a restatement of the conclusion. Theorem 2.15 is not defined in terms of the target theorem, and no fitted parameter is renamed as a prediction. The 'if' direction of Theorem 2.12 is supported by Lemma 2.8 together with the fact that a solution y1 = exp z with z' algebraic over K is genuinely constructible by generalized quadratures over K, so this is not self-definitional. The paper itself flags its real weakness: Theorem 2.15 'plays a crucial role' and is presented without proof, with the author saying its modern proof will be given in a separate paper (Section 2.4.3). That is an unproved assumption and a correctness risk, but it is not circular. Section 3 is explicitly an outline with 'basically no proofs', and its many citations to [Kho14], [Kho18a], [Kho18b], and [Kho70, Kho71] are citations to real published work, not to an unverified uniqueness theorem used to forbid alternatives. Self-citation alone does not constitute circularity here. Overall the derivation is self-contained except for the unproved black-box Theorem 2.15, which is a missing proof rather than a circular reduction.

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

No free parameters or invented entities appear: this is pure mathematics. The main assumed input is the unproved generalized Newton theorem and, in the survey part, a body of results from [Kho14] that the paper presents without proof.

assumptions (4)

domain assumption Theorem 2.15, the generalized Newton theorem: an algebraic function over a differential field of meromorphic functions has convergent Puiseux expansions at infinity with coefficients in a finite extension KP of K, uniformly over a punctured Riemann surface.
Stated in Section 2.4.3 without proof; the author defers the proof to a separate paper. This is the key tool used to prove Theorems 2.18, 2.19, and ultimately Theorem 2.12.
domain assumption Stability of the class of S-functions and the monodromy-class stability theorems from [Kho14] used throughout Section 3.3.
Theorems 3.10 through 3.14 are cited from the author's book and not proved in this paper; they underlie the strong non-representability and solvability criteria for S-functions.
standard math Frobenius theorem and Lie-Kolchin theorem used to connect monodromy group solvability to representability for Fuchsian equations.
Invoked in Section 3.3.12 as standard results from the theory of Fuchsian differential equations and linear algebraic groups.
standard math Jordan's theorem identifying the Galois group of an algebraic equation with its monodromy group.
Used in Section 3.2 to translate solvability by radicals into solvability of the monodromy group.

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Pith. "Pith review of Comments on J. F. Ritt's book "Integration in Finite Terms"." pith.science (2026). https://pith.science/paper/JHDJTGPV

@misc{pith2026190802048,
  author       = {Pith},
  title        = {Pith review of: Comments on J. F. Ritt's book "Integration in Finite Terms"},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JHDJTGPV}},
  note         = {Machine review of arXiv:1908.02048}
}

read the original abstract

The First and Second Liouville's Theorems provide correspondingly criterium for integrability of elementary functions "in finite terms" and criterium for solvability of second order linear differential equations by quadratures. The brilliant book of J.F.~Ritt contains proofs of these theorems and many other interesting results. This paper was written as comments on the book but one can read it independently. The first part of the paper contains modern proofs of The First Theorem and of a generalization of the Second Theorem for linear differential equations of any order. In the second part of the paper we present an outline of topological Galois theory which provides an alternative approach to the problem of solvability of equations in finite terms. The first section of this part deals with a topological approach to representability of algebraic functions by radicals and to the 13-th Hilbert problem. This section is written with all proofs. Next sections contain only statements of results and comments on them (basically no proofs are presented there).

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

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[2]

Burda and A

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Integrability in finite terms and actions of Lie groups

Askold Khovanskii. Integrability in finite terms and actions of Lie groups. Uspehi Mat. Nauk , 2018. To appear

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Pith tools

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

[1] [1]

V. B. Alekseev. Abel's theorem in problems and solutions . Kluwer Academic Publishers, Dordrecht, 2004. Based on the lectures of Professor V. I. Arnold, With a preface and an appendix by Arnold and an appendix by A. Khovanskii

work page 2004

[2] [2]

Burda and A

Y. Burda and A. Khovanskii. Polynomials invertible in k -radicals. Arnold Math. J. , 2(1):121--138, 2016

work page 2016

[3] [3]

A. G. Khovanski . The representability of algebroidal functions by superpositions of analytic functions and of algebroidal functions of one variable. Funkcional. Anal. i Prilo z en. , 4(2):74--79, 1970

work page 1970

[4] [4]

A. G. Khovanski . Superpositions of holomorphic functions with radicals. Uspehi Mat. Nauk , 26(3(159)):213--214, 1971

work page 1971

[5] [5]

A. G. Khovanski . Fewnomials , volume 88 of Translations of Mathematical Monographs . American Mathematical Society, Providence, RI, 1991. Translated from the Russian by Smilka Zdravkovska

work page 1991

[6] [6]

Topological G alois theory

Askold Khovanskii. Topological G alois theory . Springer Monographs in Mathematics. Springer, Heidelberg, 2014. Solvability and unsolvability of equations in finite terms, Appendices C and D by Khovanskii and Yuri Burda [Yura Burda on title page verso], Translated from the Russian by V. Timorin and V. Kirichenko (Chapters 1--7) and Lucy Kadets (Appendices...

work page 2014

[7] [7]

Integrability in finite terms and actions of Lie groups

Askold Khovanskii. Integrability in finite terms and actions of Lie groups. Uspehi Mat. Nauk , 2018. To appear

work page 2018

[8] [8]

Solvability of equations by quadratures and N ewton's theorem

Askold Khovanskii. Solvability of equations by quadratures and N ewton's theorem. Arnold Math. J. , 4(2):193--211, 2018

work page 2018