{"id":"3ea8ca6d-cf3b-4224-a95d-673e3a33fd9b","arxiv_id":"2505.24752","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For linearly reductive finite group schemes, invariant rings are generated in degree at most the order of the group scheme, while for α_q the required degree is unbounded.","lead":"The authors prove that Emmy Noether's classical bound on the degrees of generators of invariant rings also holds for finite linearly reductive group schemes, with the bound given by the order of the group scheme. They also construct infinitesimal group schemes where no such bound exists, and they extend Molien's formula to this setting.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the proof of Theorem 3.3 is internally sound and its only external input is standard classification.","rationale":"I read the proof of Theorem 3.3 in detail. Lemma 3.1's reduction to a characteristic-zero diagonal action is valid because the invariant ring of a diagonal action is a monomial algebra, and Noether's bound over C implies every invariant monomial is a product of invariant monomials of degree at most the group order. The connected-étale split and the étale step are standard. I considered whether the classification might fail in an unexamined case, but the cited results are established and the paper only needs the split form over algebraically closed fields, where it is classical. The manuscript has minor typos and the p=0 case is left implicit, but neither affects the main theorem. Therefore I would not alter the reader's ACCEPT.","tokens_in":11547,"tokens_out":28786,"duration_ms":353328,"concrete_test":"Run a direct monomial-algebra check for the smallest non-cyclic case of Lemma 3.1: let p=2, G^0=μ_2×μ_2, choose a 5-dimensional representation with weights (1,0),(0,1),(1,1),(1,0),(0,1); enumerate by computer all invariant monomials and verify that each is a product of invariant monomials of total degree ≤4. This confirms the transfer from the Noether bound in characteristic zero for the diagonal action of (Z/2)^2 and exercises the key step of Lemma 3.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified for the central claim. The proof of Theorem 3.3 reduces to Lemma 3.1 for the multiplicative connected component and to the classical Noether bound for the étale quotient. Lemma 3.1 is sound: over an algebraically closed field, representations of G^0=∏μ_{p^{e_i}} are direct sums of characters, invariant monomials are governed by the congruences (1), and the same congruences describe the invariant monomials of ∏ Z/p^{e_i}Z acting on C^n, so the characteristic-zero Noether bound gives generation in degrees ≤|G^0|; the monomial argument then transfers directly to characteristic p. Lemma 3.2 supplies (S^{G^0})^{G^ét}=S^G, and [DK15, Thm 3.22] applies because |G^ét| is prime to p. The only external input is the standard classification of linearly reductive finite group schemes (G^0 multiplicative, G^ét order prime to p, with split connected-étale sequence), cited to [AOV08, Ha15, Ch92, Na61]. This is a literature dependency, not a gap in the proof, and I see no plausible failure mode for it. The unproved p=0 case is immediate from Lemma 2.1 and the classical Noether bound.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Noether degree bound and Molien series for finite group schemes. Its main theorem (Theorem 3.3) states that for every finite linearly reductive group scheme G over a field, the invariant ring of any finite-dimensional representation is generated in degrees at most |G|, generalizing Noether's classical bound. The proof reduces to the algebraically closed case, splits the group into its connected multiplicative part and its étale quotient, and applies the classical Noether bound for finite groups of order prime to p. The paper also constructs, for the infinitesimal group scheme α_q, representations for which the generating degree grows linearly with the number of copies, proving β(α_q)=∞ (Proposition 4.1), and it gives a Molien formula for finite linearly reductive group schemes via lifting to Witt vectors (Theorem 5.4).","tokens_in":11780,"tokens_out":25751,"duration_ms":306815,"significance":"Theorem 3.3 is a clean and natural generalization of classical results, and the proof is sound: it uses the standard classification of linearly reductive finite group schemes and reduces the key Lemma 3.1 to a monomial argument over the corresponding product of cyclic groups. Proposition 4.1 provides the first infinitesimal analogue of Richman's unboundedness result and is a valuable contribution. The Molien formula, once corrected as noted below, is also useful. The paper's reliance on external classification theorems and on [LMM21] for lifting facts is appropriate and clearly stated. The proofs are detailed and, apart from the issues in Section 5 discussed below, appear correct.","major_comments":[{"comment":"Theorem 5.4(1) as stated is false for non-algebraically closed fields. The formula 1/|G| ∑_{g∈G(k)} 1/det(1−g^{−1}t, V_k) uses only k-rational points, but for example over k=Q and n odd, μ_n has μ_n(Q)={1} while the Molien series is 1/(1−t^n); this contradicts the displayed formula and is in fact acknowledged in Example 5.6. The proof itself reduces to the algebraically closed case, so the statement should either assume k is algebraically closed or replace G(k) by G(\\bar k). The same issue appears in the definition of the abstract group G_abs: for p=0 the paper defines G_abs≅G(k), but the abstract Molien series must use geometric points, not rational points (again see Example 5.6). These two points need to be corrected for the Molien formula to be stated and applied correctly.","section":"Section 5"}],"minor_comments":[{"comment":"The condition e_i > 1 should be e_i ≥ 1: μ_p is also a finite linearly reductive group scheme of multiplicative type, and the proof works for e_i=1 as well.","section":"Lemma 3.1 (page 3)"},{"comment":"There are several typos: 'propostion' for 'proposition', 'diagramme' for 'diagram', 'linarly' for 'linearly', 'simplicitiy' for 'simplicity', and in Proposition 5.1 the polynomial ring is written as k[x_1,...,n] instead of k[x_1,...,x_n].","section":"Throughout the paper"},{"comment":"The line 'q−1 (3) = deg_{x_1,...,x_l} h_r' is confusing because (3) refers to an earlier binomial identity; it should refer to assumption (3) in the list of reductions. Please renumber or clarify.","section":"Proof of Proposition 4.1 (final display, page 9)"},{"comment":"The proof of Theorem 3.3 is written only for characteristic p>0; the p=0 case is immediate from Lemma 2.1 and the classical Noether bound, but this is not stated explicitly. A sentence in Section 3 would remove ambiguity.","section":"Theorem 3.3 and Section 3"}],"recommendation":"major_revision","confidential_remarks":"The central Noether bound (Theorem 3.3) and the α_q unboundedness result (Proposition 4.1) are sound and should be published. The main problem is in Section 5: Theorem 5.4(1) is false as written for non-algebraically closed fields, and the definition of G_abs for p=0 needs to be corrected. These are local fixes, but they affect a claimed theorem, so the paper needs a major revision. The self-citation to [LMM21] is appropriate since that work is in press and supplies the lifting machinery on which the Molien section relies."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, quick take: this is a good paper, better than the arXiv abstract suggests. The central theorem is the natural group-scheme analogue of Noether's bound, and the proof is cleaner than I expected. Lemma 3.1 is the key step: representations of G^0 = product of mu_{p^{e_i}} are diagonalized, invariant monomials are governed by the congruences (1), and those same congruences describe invariants of the corresponding finite abelian group over C, so the characteristic-zero Noether bound transfers. Lemma 3.2 then hands you the etale quotient, and DK15's bound applies because |G^et| is prime to p. The reduction to algebraically closed field via Lemma 2.1 is standard. I don't see a gap in Theorem 3.3. The reliance on the classification of linearly reductive finite group schemes is real, but [AOV08, Ha15, Ch92, Na61] is established literature, not a hidden assumption. The p=0 case is immediate from Lemma 2.1 plus the classical bound, so the theorem covers all characteristics.\n\nSecond, the alpha_q example in Proposition 4.1 is the right analogue of Richman's result. The proof is long and binomial-heavy; I did not re-derive every Lucas's theorem step, but the induction has the right shape and the conclusion beta(alpha_q, rho) >= l is consistent with Richman. This is a genuine counterexample showing beta = infinity for an infinitesimal group scheme, which was missing in the literature.\n\nThird, the Molien formula is a nice bonus. It relies on LMM21's canonical lift and specialisation map, which are not yet published; that is a dependency to watch, not a flaw. Example 5.6 makes the point that the Molien series is not a function of k-rational points, and Example 5.7 shows beta can change under lifting while the Molien series does not.\n\nSoft spots: Proposition 4.1 is hard to check line-by-line; there are minor typos (e.g., 'propostion', 'SG0' for 'SG', 'x1,...,n' in 5.1), and the proof of Proposition 5.1 is compressed. None of this touches the central claim. The paper is honest about what is imported and does not oversell the results.\n\nWho it's for: people working in invariant theory, quotient singularities, and finite group schemes. It deserves a serious referee. My recommendation: send it to review; the referee should verify the binomial manipulations carefully and check LMM21's status, but the main theorem is solid.","headline":"Solid paper: Noether's bound survives for linearly reductive finite group schemes, with a Richman-style unboundedness example for alpha_q and a Molien formula; referee it.","tokens_in":739,"tokens_out":1849,"would_cite":true,"duration_ms":39201,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13A50","14L24","14L15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every linearly reductive finite group scheme $G$, the invariant ring of any representation is generated in degrees at most $|G|$.","keywords":["invariant theory","finite group schemes","linearly reductive group schemes","Noether degree bound","infinitesimal group schemes","Molien formula","positive characteristic","Witt vectors"],"falsifier":"Find a finite linearly reductive group scheme $G$ and a finite-dimensional representation $V$ for which $S(V)^G$ contains a homogeneous invariant of degree $>|G|$ that is not in the subalgebra generated by invariants of degree $\\le |G|$; such a pair would disprove Theorem 3.3. Equivalently, one could look for a linearly reductive $G$ whose abstract decomposition into $G^0 \\rtimes G^{\\mathrm{\\'et}}$ fails over some field, since the proof's reduction depends on that structure.","tokens_in":11344,"feed_emoji":"🧮","tokens_out":8998,"duration_ms":101849,"temperature":0.7,"pith_summary":"Noether's classical degree bound says that for a finite group acting on a polynomial ring in characteristic zero, the invariant ring is generated by invariants of degree at most the group order. This paper establishes the same bound for every linearly reductive finite group scheme: if $G$ is such a scheme, then $\\beta(G) \\le |G|$, meaning that for every finite-dimensional representation $V$, the invariant ring $S(V)^G$ is generated by homogeneous invariants of degree at most $|G|$. Because linear reductivity is the scheme-theoretic analogue of the condition that the group order is prime to the characteristic, this is the correct positive-characteristic generalization of Noether's theorem. The paper also shows that the infinitesimal group scheme $\\alpha_q$ has $\\beta(\\alpha_q)=\\infty$, so without linear reductivity no bound depending only on the group can exist, and it derives a Molien formula for linearly reductive finite group schemes via a lift to Witt vectors.","feed_headline":"Noether's degree bound survives for linearly reductive group schemes","feed_subtitle":"Invariant rings are generated in degree at most the group order, while α_q shows the bound fails without linear reductivity.","key_machinery":"The object that carries the proof is the Noether number $\\beta(G)$, defined as the supremum over finite-dimensional representations $\\rho$ of the minimal degree $m$ such that $S(V)^G$ is generated by homogeneous invariants of degree at most $m$. The mechanism is the structural decomposition of a linearly reductive finite group scheme $G$ as $G^0 \\rtimes G^{\\mathrm{\\'et}}$, where $G^0$ is a product of $\\mu_{p^{e_i}}$ and $G^{\\mathrm{\\'et}}$ has order prime to $p$; invariant rings are computed in two stages as $S(V)^G=(S(V)^{G^0})^{G^{\\mathrm{\\'et}}}$, with the first stage controlled by monomial invariants of a diagonalizable action and the second by the classical Noether bound for the finite group $G^{\\mathrm{\\'et}}(k)$. For the unbounded examples, the device is an explicit family of invariant polynomials $g_l$ in the $\\alpha_q$-module $lV_2$ whose degree is forced to be at least $l(q-1)$ by a degree-counting argument.","core_discovery":"The paper's central claim is Theorem 3.3: for a linearly reductive finite group scheme $G$ over any field, $\\beta(G) \\le |G|$. The proof reduces to algebraically closed fields and uses the classification of such group schemes, which writes $G$ as a semidirect product $G^0 \\rtimes G^{\\mathrm{\\'et}}$, where $G^0$ is a product of multiplicative group schemes $\\mu_{p^{e_i}}$ and $G^{\\mathrm{\\'et}}$ has order prime to $p$. Invariants for $G^0$ are monomial invariants under a diagonal action, and the Noether bound for the finite group $\\mathbb{Z}/p^{e_1}\\mathbb{Z}\\times\\cdots\\times\\mathbb{Z}/p^{e_s}\\mathbb{Z}$ bounds the degrees of their generators; invariants for the \\'etale part are then bounded by the classical Noether bound for a finite group of order prime to $p$. In the opposite direction, the paper constructs, for each $q=p^e$ and each $l$, a representation of $\\alpha_q$ on $2l$ variables whose invariant ring contains a homogeneous invariant of degree $l(q-1)$ that cannot be generated by lower-degree invariants, so $\\beta(\\alpha_q)=\\infty$. Finally, using a flat lift to the ring of Witt vectors, the paper proves a Molien formula expressing the Hilbert series of $S(V)^G$ as the classical Molien series for an associated constant group $G_{\\mathrm{abs}}$ acting on $\\mathbb{C}^n$.","pith_inferences":["A natural conjecture suggested by the paper is that every finite group scheme that is not linearly reductive has $\\beta(G)=\\infty$; the paper proves this for $\\alpha_q$ and for schemes whose \\'etale quotient is not linearly reductive, but not for mixed cases.","The Witt-vector Molien formula implies that the Hilbert series of invariants is insensitive to whether $\\beta$ is finite; Example 5.7 already shows that $\\beta$ can jump under lift while the Molien series stays the same, so one could test how far this decoupling goes by searching for linearly reductive schemes where the bound $|G|$ is attained or nearly attained.","The two-stage argument suggests a template for other classes of group schemes: decompose the group into a part whose invariants are monomial and a part governed by classical group invariants; applying the same template to group schemes whose connected part is not multiplicative could yield finite bounds under weaker hypotheses.","Because the unboundedness proof uses only Lucas's theorem and degree counts, the same construction likely gives explicit generators for the invariant rings of $\\alpha_q$-modules and could be used to compute the exact Noether number of $lV_2$ rather than only the lower bound $l$."],"forward_implications":["Every linearly reductive finite group scheme $G$ has a finite Noether number, bounded by $|G|$, so its invariant rings admit explicit finite generating sets of bounded degree.","In characteristic zero the theorem recovers Noether's original bound, since every finite group scheme is then linearly reductive and constant.","In positive characteristic the result covers diagonalizable group schemes such as $\\mu_{p^e}$ and their semidirect products with prime-to-$p$ \\'etale groups, where no such bound was previously known.","For $\\alpha_q$, the family of representations $lV_2$ forces $\\beta(\\alpha_q)=\\infty$, showing that the linearly reductive hypothesis cannot be dropped.","The Molien formula computes the Hilbert series of $S(V)^G$ from an abstract group over $\\mathbb{C}$, so the series can be obtained without computing invariants even when $G$ has no nontrivial $k$-rational points."],"supporting_citations":[{"why":"Supplies the classification of linearly reductive group schemes and the field-extension criterion used to reduce to algebraically closed fields.","marker":"[AOV08]"},{"why":"Provides the splitting $G=G^0\\rtimes G^{\\mathrm{\\'et}}$ and the description of $G^0$ as a product of $\\mu_{p^{e_i}}$ used in Lemma 3.1.","marker":"[Ha15]"},{"why":"Part of the classification of semisimple cocommutative Hopf algebras underlying the structure theorem.","marker":"[Ch92]"},{"why":"Part of the classification of linearly reductive group schemes used for the decomposition.","marker":"[Na61]"},{"why":"Gives the characteristic-zero Noether bound argument for the diagonal action of $\\mathbb{Z}/p^{e_1}\\mathbb{Z}\\times\\cdots\\times\\mathbb{Z}/p^{e_s}\\mathbb{Z}$ used in Lemma 3.1.","marker":"[Sch91]"},{"why":"Provides the Noether bound for the \\'etale quotient and the modern account of Molien's formula used in Theorem 5.4.","marker":"[DK15]"},{"why":"Gives the model unbounded example for $\\mathbb{Z}/p\\mathbb{Z}$ whose degree-reduction argument is adapted to $\\alpha_q$ in Proposition 4.1.","marker":"[Ri90]"},{"why":"Supplies the canonical flat lift to Witt vectors, the abstract group $G_{\\mathrm{abs}}$, and the specialisation map used in the Molien formula.","marker":"[LMM21]"},{"why":"The classical Molien theorem that the positive-characteristic formula is reduced to in Theorem 5.4.","marker":"[Mo97]"},{"why":"Lemma 16.3, cited for the normality statement used in Lemma 3.2(a).","marker":"[Wa79]"}],"fun_headline_variants":["Noether bound: reductive yes, α_q no","Noether bound survives for reductive group schemes; α_q fails","α_q breaks Noether bound, but reductive group schemes keep it","Linearly reductive group schemes satisfy Noether's bound","Noether's bound fails for some infinitesimal group schemes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the main bound rests on the imported classification that every linearly reductive finite group scheme splits as a semidirect product $G^0 \\rtimes G^{\\mathrm{\\'et}}$ with $G^0$ a product of $\\mu_{p^{e_i}}$ and $G^{\\mathrm{\\'et}}$ of order prime to $p$; if that classification or the splitting fails, the reduction to the two bounded cases collapses.","fun_headline_variants_meta":{"raw":{"variants":["Noether bound: reductive yes, α_q no","Noether bound survives for reductive group schemes; α_q fails","α_q breaks Noether bound, but reductive group schemes keep it","Linearly reductive group schemes satisfy Noether's bound","Noether's bound fails for some infinitesimal group schemes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001202,"raw_usage":{"total_tokens":4938,"prompt_tokens":912,"completion_tokens":4026,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":528,"completion_tokens_details":{"reasoning_tokens":3940}},"tokens_in":528,"tokens_out":4026,"duration_ms":31339,"temperature":1.0,"reasoning_tokens":3940,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:15:17.943587+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a finite linearly reductive group scheme $G$ and a finite-dimensional representation $V$ for which $S(V)^G$ contains a homogeneous invariant of degree $>|G|$ that is not in the subalgebra generated by invariants of degree $\\le |G|$; such a pair would disprove Theorem 3.3. Equivalently, one could look for a linearly reductive $G$ whose abstract decomposition into $G^0 \\rtimes G^{\\mathrm{\\'et}}$ fails over some field, since the proof's reduction depends on that structure.","supporting_citations":[],"review_version":1}