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Generic separation for modular invariants

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Pith's one-line read For modular representations of cyclic groups of prime order, generic orbits are separated and the rational invariant field is generated by one degree-p norm plus invariants of degree at most 3.

desk verdict New and credible low-degree generic separating invariants for modular cyclic groups; the field-generation argument leans on an external criterion but the stated hypotheses look right. read the letter →

arxiv 2505.20895 v1 pith:OUVHWAZ3 submitted 2025-05-27 math.RT math.AC

Fabian Reimers , Müfit Sezer This is my paper
classification math.RTmath.AC MSC 13A50
keywords modularinvarianttheoryseparatinginvariantsrationalcyclicgroupofprimeorderindecomposablerepresentationdegreeboundsgenericorbitseparation
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 attacks a problem in modular invariant theory: over a field of characteristic $p$, the invariant ring of a cyclic group of order $p$ can need generators of very high degree, but the paper shows that generic separation and the rational invariant field behave much better. For the $n$-dimensional indecomposable representation $V_n$, it constructs $n$ invariants — the linear form $x_1$, the norm $N(x_2)=x_1^{p-1}x_2-x_2^p$, and one homogeneous $f_m$ of degree $2$ or $3$ for each $3\le m\le n$ — and proves they separate every pair of orbits in the open set where the first coordinate is nonzero and generate the invariant field $k(V_n)^G$. The analogous statement for decomposable representations follows by applying the same construction to each indecomposable summand. If the proof is correct, the message is that low-degree invariants, with a single unavoidable degree-$p$ exception, capture the rational and generic information of the whole representation.

What carries the argument

The argument runs on an inductive lifting step. The projection $\phi_n:V_n\to V_{n-1}$ is $G$-equivariant, and Proposition 2 guarantees that if a set separates orbits downstairs on $B_{n-1}$, then adding one invariant that separates pairs of vectors with the same image under $\phi_n$ gives a separating set upstairs on $B_n$. The paper constructs that one invariant by solving a linear equation: for odd $n$, $f_n=x_1x_n+h$ with $h\in k[x_1,\ldots,x_{n-1}]$ and $\deg f_n=2$; for even $n$, $f_n=x_1^2x_n+h$ with $\deg f_n=3$. Existence comes from decomposing the space of degree-$2$ (or degree-$3$) monomials into weight spaces, where the weight of $x_ix_j$ is $i+j$ and of $x_ix_jx_k$ is $i+j+k$; Lemmas 1–4 show that the relevant restriction of $\Delta=\sigma-\mathrm{id}$ between weight spaces is an isomorphism, with determinant $2$ or $d-3$, so the correction terms can be solved for recursively. To conclude field generation, the paper invokes the criterion quoted as [3, Theorem 2.4], after checking that each $f_m$ has smallest positive degree in $x_m$ and that $N(x_2)$ has smallest positive degree in $x_2$.

What would settle it

For $n=4$, $p=5$, compute the extension degree $[k(x_1,x_2,x_3,x_4):k(x_1,N(x_2),f_3,f_4)]$ with $N(x_2)=x_1^4x_2-x_2^5$, $f_3=x_1x_3-\tfrac{1}{2}x_2^2+\tfrac{1}{2}x_1x_2$, and $f_4=x_1^2x_4-x_1x_2x_3+\tfrac{1}{3}x_2^3-\tfrac{1}{3}x_1^2x_2$; if the degree is $1$, the field-generation claim holds in this case, and if it is larger, the theorem is false there. This is a finite elimination computation, because $N$ is a degree-$5$ polynomial in $x_2$ over $k(x_1)$ and $f_3,f_4$ are linear in $x_3,x_4$.

Watch

Extended reading notes

Core claim

The central result is that for $G$ a cyclic group of prime order $p$ and $k$ a field of characteristic $p$, the $n$-dimensional indecomposable representation $V_n$ admits a list of exactly $n$ polynomial invariants $$x_1,\quad N(x_2)=$x_1^{{p-1}}$x_2-x_2^p,\quad f_3,\ldots,f_n,$$ where each $f_m$ is homogeneous of degree $2$ when $m$ is odd and degree $3$ when $m$ is even, such that the list separates orbits on the Zariski-open set $B_n=\{c_1\neq 0\}$ and generates the invariant field $k(V_n)^G$. The $f_m$ are built inductively with leading term $x_1x_m$ for odd $m$ and $x_1^2x_m$ for even $m$, plus a polynomial in the earlier variables. The same statement is extended to decomposable representations: with $m$ nontrivial indecomposable summands and $r$ trivial summands, there are $m+r$ linear invariants, $m$ norm invariants of degree $p$, and $n-2m-r$ invariants of degree $2$ or $3$, and these $n$ invariants generate the invariant field and separate generic orbits.

Load-bearing premise

The field-generation half of Theorem 3 rests entirely on the external criterion quoted as [3, Theorem 2.4]; the paper verifies the degree pattern the criterion asks for, but it does not re-derive that theorem, so if that criterion secretly demands a leading-coefficient or localization condition that the listed $f_m$ do not satisfy, the generating claim collapses while the separation claim still stands.

Editorial extensions

If this is right

  • The listed $n$ invariants separate any two orbits inside $\{c_1\neq 0\}$; on that open set, two points are in the same $G$-orbit if and only if all listed invariants agree.
  • For every $n\le p$, the invariant field $k(V_n)^G$ is generated by one degree-$p$ polynomial and $n-1$ polynomials of degree at most $3$.
  • For a decomposable representation, the invariant field is generated by $\dim V$ polynomials, with exactly one degree-$p$ norm per nontrivial indecomposable summand and no other generator above degree $3$.
  • Because the construction is explicit and uses only the listed invariants plus the leading-term structure, generic orbit separation does not require computing a full generating set for the invariant ring.

Reading between the lines

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

  • An implication the authors leave implicit is that the same induction should adapt to any action with a $G$-equivariant filtration by codimension-one submodules; the only new input needed at each step is a low-degree invariant whose leading term is a monomial in the new coordinate.
  • The integrality computation suggests a characteristic-independent statement the paper does not claim: after multiplication by $2$ or $d-3$, the $f_m$ are invariants for the infinite cyclic action over $\mathbb{Z}$, so reducing modulo primes that do not divide those multipliers would give analogous separating sets outside characteristic $p$.
  • If the cited lower bound is tight, the norm $N(x_2)$ is the true bottleneck for generic orbit recovery; algorithms that only use the open set $\{x_1\neq 0\}$ can evaluate one norm polynomial plus low-degree forms and skip the full invariant ring.
  • A testable extension is to compute the $f_m$ symbolically as rational functions of $p$ (with denominators $2$ and $d-3$) and specialize to all primes not dividing those denominators, yielding explicit separating lists for every relevant characteristic at once.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies generic separation and rational invariant field generation for modular representations of a cyclic group G of prime order p. For the n-dimensional indecomposable representation V_n, the authors propose the invariants x1, N(x2) = x1^{p-1}x2 - x2^p, f3, ..., fn, where each f_m is homogeneous of degree 2 for odd m and degree 3 for even m and has the form x1 x_m + h (odd m) or x1^2 x_m + h (even m) with h in k[x1, ..., x_{m-1}]. Theorems 1 and 2 prove by induction, using the equivariant projection V_n -> V_{n-1}, that these invariants separate the open set B_n = {c1 != 0}. Theorem 3 then claims that the same list is generically separating and generates the invariant field k(V)^G; the separation part is self-contained, while the field-generation part invokes [3, Theorem 2.4] after checking only that each f_m has smallest positive degree in x_m and N(x2) has smallest positive degree in x2. Theorem 4 extends the result to arbitrary finite-dimensional representations by decomposing them into indecomposable summands. The paper closes with remarks on integral invariants and a comparison with the Shank invariants from [3].

Significance. If the field-generation claim can be fully supported, the paper gives a strong and explicit low-degree bound for generic orbit separation and rational field generation in a modular setting where explicit generators are otherwise known only in small dimensions. The inductive separation method is elegant, the constructed invariants are explicit, and the proofs of the separation direction are convincing and do not rely on any fitting parameters or circular reasoning. The paper also correctly respects the lower bound of Blum-Smith et al. and, for the regular representation n = p, connects with the recent degree-3 results of Edidin and Katz. The main reservation is that the field-generation direction of Theorems 3 and 4 is not established from first principles and rests on an external criterion whose hypotheses are only partially verified in the text.

major comments (2)
  1. [Section 3.3, Theorem 3 proof (also Theorem 4 proof)] The assertion that the invariants in (3.5) generate k(V)^G rests entirely on [3, Theorem 2.4], but the paper does not state that theorem or verify its hypotheses beyond observing that each f_m is degree 1 in x_m and N(x2) is degree p in x2. A criterion for generating an invariant field from polynomials with prescribed lowest degree in a variable typically requires additional data about the leading forms or about the localization in which the induction is performed. Please quote the theorem, specify the term order or localization it uses, and check that the constructed f_m satisfy the required normal form. If the criterion indeed has no additional hypotheses, a short quotation will settle the issue; otherwise the field-generation claims in Theorem 3 and Theorem 4, and in the abstract, are unsupported.
  2. [Section 3.2, Lemma 4] The determinant computation in Lemma 4 is presented in a very compressed form: the matrix A is specified by an arrangement with ellipses and a horizontal rule, the row operation is described verbally, and the final determinant is stated as ±(d-3). Since Lemma 4 is the load-bearing step for the even-dimensional construction of f_n, please expand the computation or replace it with a more transparent argument so that the claimed isomorphism Delta_{d-1}: S'_d -> S_{d-1} can be verified. At minimum, display the matrices for the first two nontrivial cases d = 6 and d = 8 and show the determinant calculation explicitly.
minor comments (5)
  1. [Section 3.2, Proposition 4 proof] In the line 'so Delta(x1x2n - gn+2) is in Sn-1 + Sn' and in the subsequent display, the expression 'x1x2n' should read 'x1^2 x_n'; please correct this typo throughout the proof.
  2. [Section 3.2, Lemma 3] The basis descriptions (B1) and (B2) are ambiguous for small d; for instance, when d = 7 the listed monomials appear to contain duplicates. Please state the indexing convention explicitly, for example by requiring nondecreasing exponent sequences, and give the small exceptional cases separately.
  3. [Section 3.2, Lemma 3] The displayed matrix for the restriction of Delta_{d-1} to the subspace cS_d contains a '*' entry and an isolated '0' whose positions relative to the ordered bases are not explained; please clarify the matrix layout so the triangular structure and diagonal entries can be checked.
  4. [Section 3.3, Remark 1] There is a duplicated word in the sentence 'They were were constructed in [3, Theorem 2.3]'; please remove the repetition.
  5. [Section 2, Proposition 1] In the proof of Proposition 1, the step 'and hence v and w are in the same G-orbit' assumes that the invariant ring separates orbits. This is standard over algebraically closed fields, but since the proposition is stated for arbitrary infinite k, a brief justification or reference for this separation property would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the invariant construction and the generic-separation induction are self-contained, and the field-generation claim rests on an external theorem by Campbell-Chuai rather than on the paper's own inputs.

full rationale

The derivation chain is not circular. The generic-separation claim of Theorem 3 is self-contained: Proposition 2 is proved in full in the text, Theorems 1 and 2 lift B_{m-1}-separating sets to B_m-separating sets using the constructed forms f_m = x1*x_m + h (odd m) and f_m = x1^2*x_m + h (even m), and the n = 2 base case rests on the standard fact k[V_2]^G = k[x1, N(x2)]. The invariants f_m are produced by an explicit elimination algorithm (Lemmas 1-4) that solves the equation Delta(f) = 0 for a prescribed leading term; they are not defined in terms of the orbits they later separate, so the separation statement is a genuine theorem about them rather than a tautology. The field-generation half of Theorem 3 and Theorem 4 is not derived from the paper's own premises: it invokes the external criterion [3, Theorem 2.4] (Campbell-Chuai), whose authors do not overlap with the present paper, and verifies the 'smallest positive degree in x_m' hypothesis. Whether that check satisfies every hypothesis of [3, Theorem 2.4] is a rigor or correctness question, not a circularity, because the criterion's assumptions do not include the target result. The two self-citations ([9] Kohls-Sezer and [11] Sezer) are contextual: Proposition 2 is a proved generalization of [9, Theorem 1.1], and [11] appears only in the introduction as background, so neither carries the load of the main claims. Lemma 4's determinant computation is sketched, but the matrix is displayed and the resulting determinant +/- (d-3) != 0 in characteristic p is checkable. There is no fitted input, no self-definition, and no uniqueness imported from the authors' own prior work.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central construction has no fitted parameters and introduces no new mathematical entities. It depends on the external Campbell-Chuai criterion and on standard structure theory of modular representations of cyclic p-groups.

assumptions (3)
  • standard math Campbell-Chuai [3, Theorem 2.4]: a triangular system of polynomial invariants, with each invariant of smallest positive degree in its leading variable, generates the invariant field.
    Invoked in Section 3.3 (proof of Theorem 3) and in Theorem 4 to pass from the degree condition on f_m and N(x2) to field generation; the theorem is not proved or stated in detail in this paper.
  • domain assumption For the two-dimensional indecomposable representation V_2, the invariant ring is generated as a k-algebra by x1 and N(x2)=x1^{p-1}x2-x2^p.
    Base case of the induction in Theorem 3, stated in Section 3.3 and treated as standard modular invariant theory.
  • standard math The invariant field k(V)^G is the field of fractions of the invariant ring k[V]^G.
    Used in Proposition 1, with reference to Derksen-Kemper [5, Lemma 3.11.3].

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Cite this review

Pith. "Pith review of Generic separation for modular invariants." pith.science (2026). https://pith.science/paper/OUVHWAZ3

@misc{pith2026250520895,
  author       = {Pith},
  title        = {Pith review of: Generic separation for modular invariants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OUVHWAZ3}},
  note         = {Machine review of arXiv:2505.20895}
}
abstract

For modular indecomposable representations of a cyclic group $G$ of prime order $p$ we propose a list of polynomial invariants of degree $\leq 3$ that, together with a simple invariant of degree $p$, separate generic orbits and generate the field of rational invariants. A similar result is proven for decomposable representations of $G$.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Generic orbits, normal bases, and generation degree for fields of rational invariants

    math.AC 2025-06 unverdicted novelty 7.0 of 10

    Theorem: for faithful finite group actions in characteristic not dividing the group order, β_field ≤ 2D_span + 1, and equality is attained.

Reference graph

Works this paper leans on

13 extracted references · 11 canonical work pages · cited by 1 Pith paper

  1. [2]

    Blum-Smith, T

    B. Blum-Smith, T. Garcia, R. Hidalgo, and C. Rodriguez, Degree bounds for fields of rational invariants of Z/pZ and other finite groups , Journal of Pure and Applied Algebra, 228 (2024), p. 107693

    work page 2024

  2. [3]

    H. E. A. Campbell and J. Chuai, Invariant fields and localized invariant rings of p-groups, Q. J. Math., 58 (2007), pp. 151–157

    work page 2007

  3. [1]

    A. S. Bandeira, B. Blum-Smith, J. Kileel, J. Niles-Weed, A. Perry, and A. S. Wein, Estimation under group actions: Recovering orbits from invariants , Applied and Computa- tional Harmonic Analysis, 66 (2023), pp. 236–319

    work page 2023

  4. [4]

    H. E. A. Campbell and D. L. Wehlau, Modular Invariant Theory, vol. 139, Springer Science & Business Media, 2011. 1In [12] they are actually called rational invariants, which we avoid for possible confusion with elements of the invariant field k(V )G. GENERIC SEPARATION FOR MODULAR INV ARIANTS 11

    work page 2011

  5. [5]

    Derksen and G

    H. Derksen and G. Kemper, Computational Invariant Theory , vol. 130 of Encyclopaedia of Mathematical Sciences, Springer, Heidelberg, enlarged ed., 2015. With two appendices by Vladimir L. Popov, and an addendum by Norbert A’Campo and Popov, Invariant Theory and Algebraic Transformation Groups, VIII

    work page 2015

  6. [6]

    Edidin and J

    D. Edidin and J. Katz, Generic orbit recovery from invariants of very low degree , preprint, arXiv:2408.09599v2, (2024)

    arXiv 2024

  7. [7]

    Edidin and J

    D. Edidin and J. Katz, Orbit recovery from invariants of low degree in representations of finite groups, preprint, arXiv:2503.00009, (2025)

    arXiv 2025

  8. [8]

    Fleischmann, G

    P. Fleischmann, G. Kemper, and C. Woodcock, Homomorphisms, localizations and a new algorithm to construct invariant rings of finite groups , Journal of Algebra, 309 (2007), pp. 497–517. Computational Algebra

    work page 2007

Show all 13 references
  1. [9]

    Kohls and M

    M. Kohls and M. Sezer, Separating invariants for the Klein four group and cyclic groups , Internat. J. Math., 24 (2013), pp. 1350046, 11

    2013

  2. [10]

    V. L. Popov and E. B. Vinberg, Invariant theory, in Algebraic Geometry IV: Linear Alge- braic Groups Invariant Theory, Springer, 1994, pp. 123–278

    1994

  3. [11]

    Sezer, Constructing modular separating invariants , J

    M. Sezer, Constructing modular separating invariants , J. Algebra, 322 (2009), pp. 4099– 4104

    2009

  4. [12]

    R. J. Shank, S.A.G.B.I. bases for rings of formal modular seminvariants [semi-invariants] , Comment. Math. Helv., 73 (1998), pp. 548–565

    1998

  5. [13]

    D. L. Wehlau, Invariants for the modular cyclic group of prime order via classical invariant theory, J. Eur. Math. Soc. (JEMS), 15 (2013), pp. 775–803. Technische Universit¨at M¨unchen, Department of Mathematics, Boltzmannstr. 3, 85748 Garching, Germany Email address : reimers...

    2013

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