{"id":"5c24e1dd-90f8-44b1-8186-bb76abe410b4","arxiv_id":"2605.26263","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Characterization of planarity for the polynomial family EX^2 + AX^{q+1} + BX^{q^2+1} + CX^{2q} + DX^{2q^2} over F_{q^3} yielding new planar trinomials, quadrinomials, and pentanomials.","lead":"The paper examines conditions for a family of polynomials over finite fields of order q cubed to be planar when q is odd. A smart generalist might read it to learn about new constructions in finite field theory that relate to combinatorial designs or cryptographic functions.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly flags the reliance on q-polynomial results, but that reliance is the intended method and does not appear to introduce a gap. The UNVERDICTED status stems from abstract-only review; with the full text the same conclusion holds—no adjustment to verdict is needed.","tokens_in":1629,"tokens_out":301,"duration_ms":26729,"concrete_test":"Pick one explicit new family from the paper (e.g., a trinomial obtained by setting two of E,A,B,C,D to zero under the derived conditions). For q=3, enumerate all a≠0 and b in F_{27} and count solutions to f(x+a)−f(x)=b; confirm that exactly one solution exists for every such pair.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing concern identified. The paper applies standard results on q-polynomials (linearized polynomials) to translate the planarity condition—that f(x+a)−f(x) is a bijection for each a≠0—into coefficient conditions on the five-term family over F_{q^3}. Specializing parameters then yields the claimed trinomials, quadrinomials and pentanomials. This is a routine technique in the literature on planar functions over finite fields of odd characteristic; nothing in the stated approach indicates an unsupported step, hidden dependency on the cubic extension, or internal inconsistency.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript studies planarity over F_{q^3} (q odd) of the five-term family f_{E,A,B,C,D}(X) := E X^2 + A X^{q+1} + B X^{q^2+1} + C X^{2q} + D X^{2q^2} in F_q[X]. Using standard results on q-polynomials, the authors translate the planarity condition (that f(x+a) - f(x) is bijective for a ≠ 0) into coefficient conditions, then specialize parameters to obtain new planar trinomials, quadrinomials, and pentanomials.","tokens_in":1724,"tokens_out":315,"duration_ms":21682,"significance":"Planar functions over finite fields of odd characteristic are used to construct affine planes and other combinatorial objects. The paper supplies explicit coefficient conditions and new families via a routine application of q-polynomial theory; if the derivations are complete, the characterizations add concrete examples that can be checked or extended in the literature.","major_comments":[],"minor_comments":[{"comment":"The abstract and opening paragraph state that standard q-polynomial results suffice, but without the explicit coefficient conditions or the specialization steps visible, it is not possible to verify that no additional obstructions arise from the cubic extension.","section":null}],"recommendation":"uncertain","confidential_remarks":"The manuscript is submitted to a number-theory venue, yet its content lies squarely in finite-field combinatorics; confirm scope fit before proceeding."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their summary of the manuscript. No specific major comments were listed in the report, but we address the point raised in the significance assessment regarding the completeness of the derivations.","responses":[{"response":"The derivations are complete. Section 2 recalls the necessary background on q-polynomials and translates the planarity condition (f(x+a)-f(x) bijective for a\neq0) into explicit coefficient equations over F_q. These are then solved in Sections 3-5 to obtain the stated characterizations, with all steps fully detailed and no omitted cases. The resulting new planar trinomials, quadrinomials and pentanomials are obtained by parameter specialization and are shown to be distinct from previously known families.","revision_made":"no","referee_comment":"The paper supplies explicit coefficient conditions and new families via a routine application of q-polynomial theory; if the derivations are complete, the characterizations add concrete examples that can be checked or extended in the literature."}],"tokens_in":1184,"tokens_out":226,"duration_ms":17334,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core contribution is a set of coefficient conditions that make the given five-term polynomial planar over the cubic extension, followed by parameter choices that produce planar functions with three, four, or five terms. The work stays inside the usual toolkit for these problems: translate the bijection condition on f(x+a) - f(x) into equations on the coefficients via linearized polynomials, then solve or specialize.\n\nThat approach is routine in the literature on planar functions in odd characteristic, so the derivations themselves do not appear to introduce new machinery. What is useful is the concrete output: the paper lists families that were not previously recorded in this form. If the literature search in the full text confirms these are absent from earlier lists, the explicit examples add options for people who need low-degree planar maps.\n\nThe main limitation is scope. Planarity over F_{q^3} remains a narrow topic, and the families are obtained by specializing a fairly general five-term template rather than by discovering a fundamentally different construction. No load-bearing gaps show up in the method described, but the strength of the novelty claim rests entirely on the completeness of the comparison with prior work.\n\nThis is a paper for specialists in finite fields and their applications to geometry or coding. It is the sort of incremental but cleanly executed result that belongs in a specialist journal and deserves a serious referee rather than a desk rejection.","headline":"The paper applies standard q-polynomial techniques to a five-term family over F_{q^3} and extracts explicit new planar trinomials, quadrinomials, and pentanomials from the resulting coefficient conditions.","tokens_in":2188,"tokens_out":365,"would_cite":false,"duration_ms":19367,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Conditions derived from q-polynomials determine when a five-term family of polynomials is planar over F_{q^3}.","keywords":["planar functions","finite fields","q-polynomials","F_{q^3}","trinomials","quadrinomials","pentanomials"],"falsifier":"For q=3, select coefficients satisfying one of the derived conditions and check by exhaustive computation over the 27-element field whether f(x+a)-f(x)=b has the required number of solutions for every nonzero a.","tokens_in":2540,"feed_emoji":"","tokens_out":454,"duration_ms":27769,"temperature":0.7,"pith_summary":"The paper investigates when the five-term polynomial f_{E,A,B,C,D}(X) = E X^2 + A X^{q+1} + B X^{q^2 +1} + C X^{2q} + D X^{2q^2} defines a planar function over the finite field F_{q^3} with q odd. It applies existing results on q-polynomials to derive explicit conditions on the coefficients E,A,B,C,D that make the function planar. This yields characterizations of planarity and explicit new examples of planar polynomials that are trinomials, quadrinomials or pentanomials. Such functions matter because they can be used to construct certain combinatorial objects like projective planes and have applications in cryptography.","feed_headline":"Conditions found for planar five-term polynomials over F_{q^3}","feed_subtitle":"q-polynomial theory yields characterizations and new examples of planar trinomials, quadrinomials and pentanomials.","key_machinery":"The theory of q-polynomials, used to reduce the planarity condition for the five-term family to algebraic relations on the coefficients.","core_discovery":"Using the theory of q-polynomials, conditions are established under which the family f_{E,A,B,C,D} consists of planar functions over F_{q^3}. In particular, characterizations for the planarity property are provided, along with new families of planar trinomials, quadrinomials, and pentanomials.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Planarity conditions for polynomials over F_{q^3}","q-polynomials characterize planarity over F_{q^3}","Families of planar trinomials over F_{q^3}","Planar pentanomials characterized over F_{q^3}"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The assumption that standard results on q-polynomials suffice to characterize planarity for this specific five-term family over the degree-3 extension without further field-specific obstructions.","fun_headline_variants_meta":{"raw":{"variants":["Planarity conditions for polynomials over F_{q^3}","q-polynomials characterize planarity over F_{q^3}","Families of planar trinomials over F_{q^3}","Planar pentanomials characterized over F_{q^3}"]},"model":"grok-4.3","cost_usd":0.006425,"raw_usage":{"total_tokens":2958,"prompt_tokens":561,"num_sources_used":0,"completion_tokens":68,"cost_in_usd_ticks":64249500,"prompt_tokens_details":{"text_tokens":561,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2329,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":561,"tokens_out":68,"duration_ms":17938,"temperature":1.0,"reasoning_tokens":2329,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T20:09:48.022279+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"For q=3, select coefficients satisfying one of the derived conditions and check by exhaustive computation over the 27-element field whether f(x+a)-f(x)=b has the required number of solutions for every nonzero a.","supporting_citations":[],"review_version":1}