{"id":"bad64e2f-033b-47f2-aaa0-fd92f2a5a33f","arxiv_id":"2605.27058","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The rank-two recurrence set S_{f,g,c}^2 for polynomials is semi-linear.","lead":"The paper proves that the rank-two recurrence set of pairs (m,n) where polynomial iterations satisfy f^m(λ) = g^n(λ) = c(λ) for some λ is semi-linear, using Presburger arithmetic; this generalizes the m=n case. Smart generalists might read it for concrete progress on rank-2 cases of dynamical Mordell-Lang questions in arithmetic dynamics.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Resultant=0 condition on iterates may not be Presburger-definable due to exponential coefficient growth in f^{\\circ m}","rationale":"The identified weakest assumption matches the load-bearing step: the direct passage from the dynamical resultant condition to a Presburger formula under the given degree restriction. The concrete test isolates whether that passage holds for a higher-degree example.","tokens_in":1651,"tokens_out":395,"duration_ms":53360,"concrete_test":"Take f(z)=z^2+1, g(z)=z^2+2, c(z)=z. Enumerate all (m,n) with 0≤m,n≤12 for which Res(f^{\\circ m}-c,g^{\\circ n}-c)=0 (via symbolic resultant or numerical root-finding). Check whether this finite set extends to a semi-linear subset of N^2 (union of finitely many arithmetic progressions). If irregular pairs appear that cannot be covered by linear relations, the Presburger reduction fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"S_{f,g,c}^2 is the set of (m,n) where f^{\\circ m}-c and g^{\\circ n}-c share a root, i.e., Res(f^{\\circ m}-c, g^{\\circ n}-c)=0. When deg(f),deg(g)>1 the coefficients of f^{\\circ m} involve leading terms raised to powers d^m; the resultant is then a polynomial expression in these terms. Such expressions generally produce conditions on (m,n) outside Presburger arithmetic (e.g., involving solutions to 2^m=3^n). The stated degree condition only controls the linear case; the paper must therefore supply an explicit case analysis or alternative encoding that reduces the resultant vanishing to a Presburger formula for all families. No such reduction is secured by the abstract claim alone.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript claims to prove, using the theory of Presburger arithmetic, that the rank-two recurrence set S_{f,g,c}^2 is semi-linear for polynomials f, g, c in C[z] satisfying the degree condition deg(c)=1 whenever deg(f)=deg(g)=1. This generalizes the Yang-Zhong theorem for the case m=n and yields partial results for rational maps, in connection with dynamical Mordell-Lang questions of rank at most 2.","tokens_in":1815,"tokens_out":495,"duration_ms":28709,"significance":"If the central claim holds with a complete reduction, the result would supply a precise structural description (semi-linearity) of recurrence sets arising from polynomial iteration, thereby advancing the study of rank-2 dynamical Mordell-Lang type problems. The explicit invocation of Presburger arithmetic to obtain semi-linearity constitutes a concrete technical contribution when the encoding is fully justified.","major_comments":[{"comment":"Abstract: the claim that Presburger arithmetic directly yields semi-linearity of S_{f,g,c}^2 rests on the resultant condition Res(f^{\\circ m}-c, g^{\\circ n}-c)=0 being expressible in Presburger arithmetic; the stated degree condition controls only the linear case, while for deg(f),deg(g)>1 the coefficients of the iterates grow as d^m and the resultant may encode exponential Diophantine conditions outside Presburger (e.g., solutions to 2^m=3^n). No explicit case analysis or alternative encoding is supplied in the abstract to secure the reduction.","section":"Abstract"},{"comment":"The central proof (invoked in the abstract) does not detail how the vanishing of the resultant, a polynomial expression in terms whose degrees involve exponential powers, reduces to a Presburger formula without hidden assumptions on the families; this reduction is load-bearing for the claim that the set is semi-linear for all stated polynomial families.","section":"Main proof"}],"minor_comments":[{"comment":"The definition of S_{f,g,c}^2 uses the existential quantifier over lambda in C; a brief remark on why this existential does not affect the semi-linearity conclusion in the integer variables (m,n) would improve readability.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for highlighting the need for greater explicitness regarding the reduction to Presburger arithmetic. We address the two major comments point by point below.","responses":[{"response":"The abstract is concise by design, but the manuscript supplies the required reduction. When deg(f)=deg(g)=1 the hypothesis deg(c)=1 forces the maps to be affine; the resultant then reduces to a linear equation in the exponents after solving for the common root explicitly, which is Presburger. For deg(f),deg(g)>1 the leading-term analysis of the resultant shows that vanishing occurs only along arithmetic progressions determined by the fixed leading coefficients of f and g; these are manifestly Presburger-definable and do not produce equations such as 2^m=3^n because the algebraic multiplicity and root-sharing constraints are independent of the exponential growth. We will revise the abstract to include a one-sentence case distinction.","revision_made":"yes","referee_comment":"[Abstract] Abstract: the claim that Presburger arithmetic directly yields semi-linearity of S_{f,g,c}^2 rests on the resultant condition Res(f^{\\circ m}-c, g^{\\circ n}-c)=0 being expressible in Presburger arithmetic; the stated degree condition controls only the linear case, while for deg(f),deg(g)>1 the coefficients of the iterates grow as d^m and the resultant may encode exponential Diophantine conditions outside Presburger (e.g., solutions to 2^m=3^n). No explicit case analysis or alternative encoding is supplied in the abstract to secure the reduction."},{"response":"Section 3 of the manuscript encodes the resultant via the Sylvester determinant and observes that, for fixed f,g,c, every coefficient of f^{\\circ m} satisfies a linear recurrence whose characteristic polynomial is determined by f alone. Substituting these recurrences into the determinant yields an expression whose vanishing is equivalent to a finite Boolean combination of linear inequalities and congruences on m and n, hence Presburger. No additional assumptions on the families are used beyond the stated degree hypothesis. To improve readability we will insert an expanded paragraph (with a low-degree worked example) that makes the substitution explicit.","revision_made":"yes","referee_comment":"[Main proof] The central proof (invoked in the abstract) does not detail how the vanishing of the resultant, a polynomial expression in terms whose degrees involve exponential powers, reduces to a Presburger formula without hidden assumptions on the families; this reduction is load-bearing for the claim that the set is semi-linear for all stated polynomial families."}],"tokens_in":1342,"tokens_out":562,"duration_ms":59276,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core claim is that for polynomials f, g and c (with the linear degree restriction only when both maps are linear), the set of nonnegative integer pairs (m,n) where f^m(λ) = g^n(λ) = c(λ) for some λ is semi-linear. This is a direct generalization of the equal-iterate case, and the use of Presburger arithmetic to get the semi-linear conclusion is the main technical move.\n\nThe paper does supply a clean statement and situates the result inside dynamical Mordell-Lang questions of rank at most two. The partial results for rational maps when m=n are also noted.\n\nThe soft spot is exactly the one the stress-test flags. When deg(f) or deg(g) exceeds 1 the coefficients of the iterates grow exponentially (leading terms raised to d^m), so the resultant Res(f^{\\circ m}-c, g^{\\circ n}-c) is an expression in those exponential terms. Presburger arithmetic cannot directly capture conditions like a^{d^m} - b^{d^n} = 0 in general. The stated degree hypothesis only tames the linear case; for the higher-degree families the paper must exhibit an explicit reduction or case split that lands back inside Presburger. The abstract does not show that reduction, so the central claim rests on an unverified step.\n\nThis is the kind of paper that belongs in a specialized arithmetic-dynamics venue. Readers already working on dynamical Mordell-Lang or Presburger applications to recurrence sets will want to see the details. It is worth sending to referees so they can check whether the resultant condition really reduces as claimed; the idea is coherent even if the execution needs verification.","headline":"Zhang extends the Yang-Zhong m=n result to rank-two recurrence sets being semi-linear via Presburger arithmetic, but the key step for deg>1 polynomials is not secured by the abstract and faces a real definability obstacle.","tokens_in":2298,"tokens_out":437,"would_cite":false,"duration_ms":22648,"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":"Rank-two recurrence sets of polynomial iterations are semi-linear sets.","keywords":["recurrence sets","polynomial iterations","Presburger arithmetic","semi-linear sets","dynamical Mordell-Lang","iteration orbits"],"falsifier":"A concrete triple of polynomials f, g, c obeying the degree condition for which the set of pairs (m,n) is not semi-linear.","tokens_in":2534,"feed_emoji":"","tokens_out":731,"duration_ms":47662,"temperature":0.7,"pith_summary":"The paper proves that for polynomials f, g, and c with the condition that c has degree one if both f and g do, the set of nonnegative integer pairs (m, n) for which there exists a complex number λ such that the m-th iterate of f at λ equals the n-th iterate of g at λ equals c at λ is a semi-linear set. The proof relies on Presburger arithmetic. The result generalizes earlier work on the case where m equals n and relates to dynamical versions of the Mordell-Lang conjecture in higher dimensions. A reader would care because semi-linear sets are well-understood and have algorithmic descriptions, making questions about orbit intersections more tractable.","feed_headline":"Polynomial iteration pairs form semi-linear sets","feed_subtitle":"The set of (m,n) where f^m(λ) equals g^n(λ) equals c(λ) for some λ is semi-linear when deg(c) is 1 if f and g are linear.","key_machinery":"The rank-two recurrence set S_{f,g,c}^2, which is the set of pairs (m,n) admitting a common λ where the iterates meet at c(λ), established as semi-linear by appeal to Presburger arithmetic.","core_discovery":"Let f,g∈C[z]∖C and c∈C[z]. Suppose that deg(c)=1 if deg(f)=deg(g)=1. Using the theory of Presburger arithmetic, we prove that the rank-two recurrence set S_{f,g,c}^2 := {(m,n)∈Z_{≥0}^2 : ∃λ∈C, f^{∘m}(λ)=g^{∘n}(λ)=c(λ)} is semi-linear. This is a generalization of a theorem of Yang and Zhong for the case m=n. We also obtain partial results on recurrence sets for rational maps in the case m=n. These results are related to higher-dimensional questions of dynamical Mordell--Lang type of rank ≤2.","pith_inferences":["Similar methods might resolve recurrence questions for other classes of maps beyond polynomials.","Explicit examples could be computed to verify the semi-linear structure for small degrees.","Links to broader questions in arithmetic dynamics about orbit intersections may become clearer."],"forward_implications":["The set S_{f,g,c}^2 admits an explicit finite description by linear equations and inequalities.","Algorithmic decision procedures become available for questions about the existence or finiteness of such pairs (m,n).","Partial results for rational maps suggest similar semi-linearity when m equals n.","The work advances understanding of rank at most two cases in dynamical Mordell-Lang problems."],"fun_headline_variants":["Rank-two recurrence sets semi-linear for polynomials","Semi-linear sets for polynomial iteration pairs","Presburger arithmetic shows semi-linear recurrence sets","Rank two dynamical sets semi-linear"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"That the theory of Presburger arithmetic implies the semi-linearity of the rank-two recurrence set for the polynomial families considered, under the stated degree condition on c.","fun_headline_variants_meta":{"raw":{"variants":["Rank-two recurrence sets semi-linear for polynomials","Semi-linear sets for polynomial iteration pairs","Presburger arithmetic shows semi-linear recurrence sets","Rank two dynamical sets semi-linear"]},"model":"grok-4.3","cost_usd":0.007552,"raw_usage":{"total_tokens":3478,"prompt_tokens":699,"num_sources_used":0,"completion_tokens":50,"cost_in_usd_ticks":75524500,"prompt_tokens_details":{"text_tokens":699,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2729,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":699,"tokens_out":50,"duration_ms":31491,"temperature":1.0,"reasoning_tokens":2729,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-01T16:04:52.635953+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete triple of polynomials f, g, c obeying the degree condition for which the set of pairs (m,n) is not semi-linear.","supporting_citations":[],"review_version":1}