{"id":"8ad39564-b404-439e-9f6e-bfd997b72cc5","arxiv_id":"2601.09415","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For an infinite family of quadrinomials over F_{q^{2t}}, new sufficient conditions are proven for being scattered, strictly generalizing prior results, and the family is shown inequivalent to known pseudoregulus and Lunardon–Polverino polynomials.","lead":"Finite fields have a small set of known 'scattered' polynomials used to build error-correcting codes; this paper widens the range of parameters for which one family of four-term polynomials is proved scattered. It also shows these polynomials are genuinely different from the three previously known infinite families.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Omitted proof of Prop. 2.4 for N=1, h²≠−1 is load-bearing; the conditional verdict should stand until the proof is supplied.","rationale":"The reader's weakest-assumption identification is correct: Proposition 2.4 is load-bearing and its extension to N(h)=1, h²≠−1 is asserted without proof. The central theorem is not jeopardized by an obvious internal inconsistency; the statement of Prop. 2.4 appears plausible and is supported by the fact that both kernel and image intersection conditions reduce to h^{q^{2s}+1}=-ε, which Prop. 2.3 excludes. However, because the omitted proof is used at the very start of both main proof branches, the paper remains conditional until that derivation (or a citation covering the N(h)=1 case) is included. The reader's CONDITIONAL verdict is therefore appropriate; my stress-test does not move it.","tokens_in":32892,"tokens_out":21205,"duration_ms":182523,"concrete_test":"For q ∈ {3,5}, t ∈ {3,5,7}, all admissible s with gcd(s,2t)=1, and all h with N_{q^{2t}/q^t}(h)=1 and h²≠−1, compute dim_{F_{qt}}(ker L_m ∩ ker M) and dim_{F_{qt}}(im L_m ∩ im M) directly from (6); also test the trace identity in Prop. 2.8. Any nonzero intersection is a counterexample. For a symbolic check, derive from (6) that each intersection condition is equivalent to h^{q^{2s}+1}=-ε and verify it is excluded by Prop. 2.3, then add a short proof of Prop. 2.8 in the N(h)=1 case via the same trace computation used for N(h)=-1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The text immediately before Prop. 2.4 states: \"Although the next results are proven in [12] for N(h)=-1, these hold for N(h)=1 and h²≠-1, as well. Since the techniques are similar, we omit the proof.\" Proposition 2.4 is the direct-sum decomposition F_{q^{2t}} = ker L_m ⊕ ker M and im L_m ⊕ im M over F_{qt}. This is used at the first step of the proofs of Theorems 3.2 and 3.4 to write every x as x1+x2 and then to equate the im L_m and im M components. Without it, the entire case split in the new N(h)=1 branch loses its foundation. Proposition 2.8, also asserted for N(h)=1 without proof, is additionally used to obtain the contradiction in Theorem 3.4, Case 3.1. This is a genuine proof gap rather than a demonstrated counterexample. A direct check from equations (6) suggests the assertion is true: writing ε=N_{q^{2t}/q^t}(h), nonzero intersection of either the kernel pair or the image pair forces h^{q^{2s}+1}=-ε, which is excluded by Prop. 2.3 for both ε=1 and ε=-1 under the stated hypotheses. But the paper should supply this reduction, or cite a source covering the N(h)=1 case, before the generalization is fully verified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies F_q-scatteredness of the q^s-linearized quadrinomial ψ_{m,h,s} over F_{q^{2t}}, and states sufficient conditions in Theorem 1.1 depending on t, q mod 4, the norm N_{q^{2t}/q^t}(h), and membership of m in the sets P_s^+, P_s^-. The proof splits into the cases N(h)=-1 and N(h)=1: it writes ψ = L_m + M, uses a direct-sum decomposition of F_{q^{2t}} into kernels and images of L_m and M, and then performs a detailed case analysis on a putative multiplier γ. The paper also claims that these results strictly include previous scatteredness results for ψ_{m,h,s}; it proves non-equivalence with pseudoregulus-type and LP-type polynomials, gives a classification of GL-equivalences among ψ_{m,h,s} in Theorem 4.3, computes stabilizers, and proposes necessary conditions and a conjecture.","tokens_in":33260,"tokens_out":12948,"duration_ms":124294,"significance":"If Theorem 1.1 is correct, this is a genuine contribution: it unifies and extends the previously known scattered quadrinomial families, gives explicit new scattered polynomials, and therefore new MRD codes. The proof is largely a transparent, parameter-free case analysis, and the paper clearly identifies the new cases. The main weakness is that a load-bearing direct-sum lemma for the new N(h)=1 case is explicitly left unproved, and another used proposition is only partially proved. The result is plausible and likely fixable, but the manuscript in its present form does not fully support the central claim in the new branch.","major_comments":[{"comment":"The text before Proposition 2.4 says that results proved in [12] for N(h)=-1 also hold for N(h)=1, h^2≠-1, 'since the techniques are similar', and omits the proof. This decomposition is used at the first step of Theorems 3.2 and 3.4, and again after Eq. (11), to equate the im L_m and im M components and to conclude x=x1=x2=0 when a=0. Since the N(h)=1 branch is exactly the genuinely new part of Theorem 1.1, a citation to [12] cannot cover it. Please supply the proof, e.g. using (6) to show that a nonzero intersection of either the kernel pair or the image pair forces h^{q^{2s}+1}=-ε, contradicting Proposition 2.3 for both ε=±1. Propositions 2.6 and 2.7 are also covered by the same omitted-proof sentence and are used in deriving (11); include their proofs as well.","section":"Section 2, Proposition 2.4"},{"comment":"Proposition 2.8 is stated for both N(h)=-1 and N(h)=1, h^2≠-1, but the proof computes only the N(h)=-1 case and ends with 'a similar argument applies' for N(h)=1. This is not a purely cosmetic variant: the norm condition changes the signs in (6), and Proposition 2.3(ii) must be invoked. The proposition is used in Theorem 3.4, Case 3.1, Eq. (33), to conclude m∈P_s^+ and obtain the contradiction. Please write out the N(h)=1 computation in full, explicitly showing the use of h^{q^{2s}+1}≠-1.","section":"Section 3.2, Proposition 2.8"}],"minor_comments":[{"comment":"The polynomial displayed in Conjecture 6.3 is not the ψ_{m,h,s} of Eq. (3): the second term has exponent X^{q^{st}-1} and the third term is X^{q^{s(t+1)}}, so the displayed object is not even q^s-linearized. The conjecture should refer to the polynomial (3).","section":"Section 6, Conjecture 6.3"},{"comment":"Lemma 2.5 is stated without proof. It is elementary, but part (ii) is used in the proofs of Theorems 3.2 and 3.4, so a short verification should be included.","section":"Section 2, Lemma 2.5"},{"comment":"The proof of Theorem 4.3 contains many compressed reductions, e.g. 'which in this case is equivalent to'. Since Theorem 4.3 supports Corollary 4.4 and the non-equivalence claims, expanding these reductions would make the argument substantially easier to verify.","section":"Appendix, Theorem 4.3"},{"comment":"There are minor typographical issues, e.g. 'the the' in Lemma 2.1(i). Also, Theorem 1.1 states (m,h)∈F_{q^t}×F_{q^{2t}}, while the definition (3) and the proof require m,h nonzero; this should be stated explicitly.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"For the editor: I do not see circularity or questionable citation practice; the issue is a genuine but likely fixable proof gap. The revision should be checked specifically on whether the N(h)=1 proofs of Proposition 2.4 and Proposition 2.8 are written out in full. If they are, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe thing to know: this is a genuine extension of the only known infinite family of maximum scattered quadrinomials, and the main theorem is coherent. The caveat is that one load-bearing lemma, Proposition 2.4, is asserted for the new N(h)=1, h^2≠-1 case without proof, and the proof of the main theorem leans on it.\n\nWhat is actually new: Theorem 1.1 covers m outside P± under norm ±1 and adds the P+ case with norm −1; it strictly contains the subcases from [1,12,13,18,22,23]. The machinery is a long case analysis, but the structure is clear: split F_{q^{2t}} via kernels/images of the two summands, reduce the scatteredness equation to systems, then use norm conditions. Theorem 4.3 and Corollary 4.4 give GL-nonequivalence criteria and actually produce new examples; the stabilizer computation in Section 5 is also useful. This is solid subfield work, not a paradigm shift.\n\nThe soft spots. Proposition 2.4 is the foundation: the proofs of Theorems 3.2 and 3.4 start by writing every x as x1+x2 and then equating components in im L_m ⊕ im M. The paper says the N=1, h^2≠−1 case follows by similar techniques and omits it. Proposition 2.8 is in the same boat. That is a genuine gap, even though I think it is fillable: the stress-test note gives a plausible direct argument from (6), and the statement is probably true. But it is load-bearing, so the authors should supply the proof or cite a source that covers N=1. The Appendix Theorem 4.3 is long coefficient-comparison algebra; it is not machine-checked, and the exposition has at least one typo in Conjecture 6.3. These are minor relative to the main theorem. The citation practice is fine; leaning on prior lemmas from overlapping authors is not circular here because the target theorem is not assumed.\n\nWho this is for: people working on scattered polynomials, MRD codes, and finite geometry. It deserves a serious referee. My recommendation: send it to peer review, with a clear request to prove Proposition 2.4 and Proposition 2.8 for N=1 (or cite a proof) before acceptance. If that lemma is fixed, the core result stands.","headline":"A solid generalization of the known scattered quadrinomial family, with the caveat that a load-bearing decomposition lemma is asserted without proof for the new N(h)=1 range.","tokens_in":33727,"tokens_out":1855,"would_cite":true,"duration_ms":17533,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11T06","11T71","94B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A quadrinomial family over F_{q^{2t}} is shown scattered under conditions strictly broader than previous results, yielding new MRD codes.","keywords":["Linearized polynomials","Scattered polynomials","MRD codes","Finite fields","Linear sets","Quadrinomials","Trace-kernel sets","Rank metric"],"falsifier":"A direct computer search for $q=3$, $t=3$, $s=1$: choose $h \\in F_{3^6} \\setminus F_{3^3}$ with $N_{3^6/3^3}(h)=1$ and $h^2 \\neq -1$, choose $m \\in F_{3^3} \\setminus (P_1^+ \\cup P_1^-)$, and compute $\\dim_{F_{3^3}}(\\ker L_m \\cap \\ker M)$. If the dimension is positive, Proposition 2.4 is false for the norm-one case; alternatively, test the scatteredness ratio directly by searching for two $F_3$-linearly independent $x,y \\in F_{3^6}$ with $\\psi_{m,h,s}(x)/x = \\psi_{m,h,s}(y)/y$, which would disprove Theorem 3.4.","tokens_in":32801,"feed_emoji":"🧮","tokens_out":8783,"duration_ms":70624,"temperature":0.7,"texified_at":"2026-08-05T20:47:44.644402+00:00","pith_summary":"The paper establishes sufficient conditions under which the four-term $q^s$-linearized polynomial $\\psi_{m,h,s}(X) = m(X^{q^s} - h^{1-q^{s(t+1)}}X^{q^{s(t+1)}}) + X^{q^{s(t-1)}} + h^{1-q^{s(2t-1)}}X^{q^{s(2t-1)}}$ is a scattered polynomial over $F_{q^{2t}}$. Scatteredness means that the ratio $\\psi(x)/x$ takes each value on at most $q$ inputs, which makes the associated rank-metric code $\\{aX + b\\psi\\}$ a maximum-rank-distance (MRD) code. The theorem splits according to the parity of $t$ and the residue of $q$ modulo 4, with conditions on $m$ lying in $F_{q^t}$ and avoiding the sets $P_s^+$ and $P_s^-$ of $(q^s+1)$-st and $(q^s-1)$-st powers of trace-zero elements, and on the norm $N_{q^{2t}/q^t}(h)$. These conditions strictly include all previously published scatteredness results for this family. A reader who accepts the proof gets a new supply of scattered polynomials and MRD codes beyond the three known infinite families.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":12374,"prompt_tokens":1171,"completion_tokens":11203,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":1171,"completion_tokens_details":{"reasoning_tokens":9934}},"feed_headline":"New scattered quadrinomials exist in F_{q^{2t}}","feed_subtitle":"Generalized conditions prove quadrinomials scattered and yield MRD codes beyond the three classical families.","key_machinery":"The proof is carried by the decomposition of $F_{q^{2t}}$ as an $F_{q^t}$-vector space into the direct sum $\\ker L_m \\oplus \\ker M$ (and $\\operatorname{im} L_m \\oplus \\operatorname{im} M$), where $L_m = m(X^{q^s} - h^{1-q^{s(t+1)}}X^{q^{s(t+1)}})$ and $M = X^{q^{s(t-1)}} + h^{1-q^{s(2t-1)}}X^{q^{s(2t-1)}}$. Two auxiliary $F_{q^t}$-linear maps $R = X^{q^{st}} + h^{q^{s(t-1)}-q^s}X$ and $T = X^{q^{st}} + h^{q^s-q^{s(t-1)}}X$ provide 1-dimensional kernels whose nonzero elements $\\rho$ and $\\tau = h^{q^{s(t-1)}-q^s}\\rho$ give bases $\\{1,\\rho\\}$ and $\\{1,\\tau\\}$ of $F_{q^{2t}}$. Writing $\\gamma$ in these bases turns the scatteredness test $\\psi(\\gamma x) = \\gamma \\psi(x)$ into a pair of equations whose coefficient contradictions force $m$ into the forbidden sets $P_s^\\pm$.","core_discovery":"The central claim, Theorem 1.1, is that for $t \\geq 3$, $q$ odd, $\\gcd(s,2t)=1$, $\\psi_{m,h,s}$ is scattered when (i) $t$ even, or $t$ odd with $q \\equiv 1 \\pmod{4}$, $m \\notin P_s^+ \\cup P_s^-$ and $N(h)=\\pm 1$; or (ii) $t$ odd, $q \\equiv 3 \\pmod{4}$, and either $m \\in P_s^+$ with $N(h)=-1$, or $m \\notin P_s^+ \\cup P_s^-$ with $N(h)=1$ and $h^2 \\neq -1$. The proof writes $\\psi = L_m + M$ and decomposes $F_{q^{2t}}$ as an $F_{q^t}$-vector space into direct sums of kernels and images of these summands, then forces the scattered condition $\\psi(\\gamma x) = \\gamma \\psi(x)$ through a $2 \\times 2$ linear system whose possible solutions contradict the assumptions on $m$. For $t \\geq 5$ the paper also proves the quadrinomials are $\\Gamma L$-equivalent neither to pseudoregulus-type monomials nor to Lunardon–Polverino binomials, and it computes the stabilize","pith_inferences":["Conjecture 6.3 asserts the converse; if true, the sufficient conditions in Theorem 1.1 are exactly the scatteredness boundary, suggesting a complete classification of scattered quadrinomials of this shape.","Because P_s^± are independent of s (Lemma 2.2), the conditions may depend only on q and t and on whether m is a (q±1)-st power of a trace-zero element; this could be checked by testing the same property for s=1 in small examples.","A testable extension is to relax gcd(s,2t)=1 or allow q even; the proof's use of parity suggests the even-q case may need separate tools, but if the same direct-sum decomposition holds there, the method would generalize.","Theorem 4.3 gives explicit necessary conditions for two quadrinomials to be GL-equivalent; counting the number of inequivalent pairs (m,h) satisfying Theorem 1.1 would quantify how much larger this family is than previously known ones."],"forward_implications":["For every triple (m,h,s) satisfying Theorem 1.1, the code {aX + bψ_{m,h,s} : a,b ∈ F_{q^{2t}}} is a linear MRD code with parameters (2t, 2t, q; 2t-1).","The families of pseudoregulus-type monomials and Lunardon–Polverino binomials are not ΓL-equivalent to ψ_{m,h,s} for t≥5, so the newly scattered polynomials are genuinely new objects.","The necessary-condition theorem shows that m∈P_s^- together with h∈F_{q^t} and N(h)=±1 forces ψ_{m,h,s} not to be scattered, settling one direction of the proposed characterization.","Since scatteredness is preserved under adjoints, the result also yields scattered adjoint polynomials and their associated linear sets.","The stabilizer computation fixes the right idealizer of the associated MRD code as F_{q^2}, an invariant that can distinguish these codes from others."],"fun_headline_variants":["Generalized scattered quadrinomials proved in F_{q^{2t}}","Quadrinomials scattered for infinite q,t beyond classical families","New scattered quadrinomials extend known families","Scattered quadrinomials: conditions and MRD codes","Generalized conditions yield scattered quadrinomials"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole proof rests on Proposition 2.4, which asserts that $F_{q^{2t}}$ is the $F_{q^t}$-direct sum of the kernels (and of the images) of $L_m$ and $M$; the paper explicitly notes that for the norm-one, $h^2 \\neq -1$ case this proposition is carried over from a companion paper by analogy, with the proof omitted, so if that extension fails, the central theorem has no foundation.","fun_headline_variants_meta":{"raw":{"variants":["Generalized scattered quadrinomials proved in F_{q^{2t}}","Quadrinomials scattered for infinite q,t beyond classical families","New scattered quadrinomials extend known families","Scattered quadrinomials: conditions and MRD codes","Generalized conditions yield scattered quadrinomials"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000803,"raw_usage":{"total_tokens":3371,"prompt_tokens":754,"completion_tokens":2617,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":2533}},"tokens_in":498,"tokens_out":2617,"duration_ms":17954,"temperature":1.0,"reasoning_tokens":2533,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T10:35:56.437678+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct computer search for $q=3$, $t=3$, $s=1$: choose $h \\in F_{3^6} \\setminus F_{3^3}$ with $N_{3^6/3^3}(h)=1$ and $h^2 \\neq -1$, choose $m \\in F_{3^3} \\setminus (P_1^+ \\cup P_1^-)$, and compute $\\dim_{F_{3^3}}(\\ker L_m \\cap \\ker M)$. If the dimension is positive, Proposition 2.4 is false for the norm-one case; alternatively, test the scatteredness ratio directly by searching for two $F_3$-linearly independent $x,y \\in F_{3^6}$ with $\\psi_{m,h,s}(x)/x = \\psi_{m,h,s}(y)/y$, which would disprove Theorem 3.4.","supporting_citations":[],"review_version":1}