{"id":"03a42707-9723-4fcc-9c22-2b39a2e594de","arxiv_id":"2507.05027","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a dominant rational self-map on a projective variety, the height associated with a closed subscheme vanishes relative to an ample height along generic orbits whenever a dynamical degree is strictly smaller than the arithmetic degree.","lead":"This paper proves an unconditional theorem about generalized greatest common divisors: for a rational self-map with enough dynamical complexity, the height that measures common divisors of coordinates in a generic orbit grows far slower than the height of the orbit itself. The result is a rigorous, hypothesis-specific step toward a conjecture that previously required Vojta's conjecture.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Morphism-case reduction in Theorem 1.5 uses h_Y≤h_Z for Z containing Y; this inequality fails, so the arbitrary-Y morphism case is unproved.","rationale":"The reader's accepted verdict identifies the cited [24, Lemma 2.4] as the weakest assumption. I agree that the external lemma is worth scrutiny, but the proof of Theorem 1.5 contains a more immediate internal gap: the reduction used to handle arbitrary Y when f is a morphism relies on an inequality between heights that is false. The height associated with a larger complete-intersection scheme containing Y can be much smaller than the height of Y, as the point-in-line example shows. Since the theorem's morphism case is stated for all proper closed subschemes, this gap is load-bearing. It does not necessarily invalidate the non-morphism case where Y is assumed pure and regularly embedded, nor does it show the theorem is false; the result may be repairable by a different reduction or by tightening the statement. For that reason I recommend conditional acceptance rather than rejection. The paper otherwise contains substantial original ideas, careful technical work, and explicit limiting examples, which deserve credit.","tokens_in":19955,"tokens_out":25294,"duration_ms":324893,"concrete_test":"Recompute the reduction step on the explicit example X=P^2, Y={0:0:1}, Z={x=0}, and P_n=(2^n:2^n:1) using the height functions of [28]. The standard formulas give h_Y(P_n)=n log 2 and h_Z(P_n)=0, falsifying the inequality h_Y ≤ h_Z+O(1) used in the proof. Then re-run the morphism-case argument with this Y and the divisors D_1,...,D_{N-l} produced by the proof's linear system to check whether the claimed bound on h_Y actually follows; if it does not, the theorem needs either a corrected reduction or a restricted hypothesis.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"At the start of the proof of Theorem 1.5 (Section 2), the case where f is a morphism is reduced to the case where Y is pure dimensional and regularly embedded. The proof chooses general divisors D_1,...,D_{N-l} containing Y and asserts h_Y ≤ h_{D_1∩...∩D_{N-l}} + O(1). For a closed subscheme Z containing Y we have I_Z ⊂ I_Y, but the asserted inequality is false in general. Example in P^2: take Y=(0:0:1), Z={x=0}, so Y⊂Z, and P_n=(2^n:2^n:1). Then h_Y(P_n)=n log 2 while h_Z(P_n)=0; hence h_Y is not bounded above by h_Z. Replacing Y by a larger complete-intersection subscheme therefore does not control h_Y. Since the theorem states the morphism case for arbitrary proper closed subschemes Y, this reduction leaves that case unproved as written. The argument would need a different reduction, for instance using divisors vanishing to high order on Y, or the theorem statement would need to be restricted to pure-dimensional regular Y even in the morphism case.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for a dominant rational self-map f on a smooth projective variety X over \\overline{\\mathbb{Q}}, a proper closed subscheme Y, and a point x with generic well-defined orbit, the generalized gcd height h_Y(f^n(x)) is asymptotically negligible compared with any ample height h_H(f^n(x)), provided that a suitable dynamical degree is strictly smaller than the arithmetic degree of x. The main theorem (Theorem 1.5) gives this for a morphism with arbitrary Y, and for non-morphisms under the additional assumptions that Y is pure dimensional, regularly embedded, and contained in the iteratively finite locus X_f^{back}. The proof combines cohomological estimates for ideals of preimages f^{-n}(Y) with a lower bound for h_H along the orbit imported from the author's earlier work [24].","tokens_in":20176,"tokens_out":17343,"duration_ms":196516,"significance":"If the proof is completed, this is a strong unconditional result: it gives a dynamical criterion under which generalized greatest common divisors along orbits grow slower than every ample height, without assuming Vojta's conjecture. The technical heart, especially Proposition 2.10 and its Segre-class estimates, is substantial and appears to be new. The paper also gives useful examples showing the necessity of the condition Y\\subset X_f^{back}. However, the morphism case of the main theorem is not proved as written because the reduction to pure-dimensional regularly embedded Y is based on a false height inequality. This must be repaired or the statement must be restricted.","major_comments":[{"comment":"The reduction of the morphism case to the case where Y is pure dimensional and regularly embedded is invalid. The proof asserts that for a general sequence D_1,...,D_{N-l} in the linear system defined by H^0(X,I\\otimes L), with Z=D_1\\cap\\cdots\\cap D_{N-l}, one has h_Y\\leq h_Z+O(1) because Y\\subset Z. This inequality is false in general. For example, take X=\\mathbb{P}^2, Y=(0:0:1), Z=\\{x=0\\}, and P_n=(2^n:2^n:1). Then h_Y(P_n)=n\\log 2+O(1), while h_Z(P_n)=O(1) (in fact 0 for the standard height). Thus h_Y is not bounded above by h_Z. In general, for Y\\subset Z the height inequality goes in the opposite direction: h_Z\\leq h_Y+O(1). Consequently the morphism case of Theorem 1.5, which allows arbitrary proper closed subschemes Y, is not established by the given argument. The theorem either needs a different reduction (for instance, controlling h_Y through h_{Y_{\\mathrm{red}}} and then handling the reduced scheme, if that is feasible) or its statement must be restricted to pure-dimensional regularly embedded Y in the morphism case as well.","section":"Section 2, proof of Theorem 1.5 (first paragraph, morphism-case reduction)"}],"minor_comments":[{"comment":"There is a duplicated phrase 'Namely, Namely' in the normalization paragraph; please fix it.","section":"Introduction, conventions"},{"comment":"The word 'infintie' should be 'infinite'. Similar typographical errors occur elsewhere, such as 'allow' for 'arrow' in Example 3.1.","section":"Lemma 2.9"},{"comment":"The theorem states that Y has dimension l, while the abstract phrases the hypothesis as d_c(f)^{1/c}<\\alpha_f(x) with c the codimension. For non-pure-dimensional Y the relation between l and c should be stated explicitly, or Y should be assumed pure dimensional in all cases.","section":"Theorem 1.5 and abstract"},{"comment":"The final lower bound for h_H along the orbit is imported from [24, Lemma 2.4]. Since this is load-bearing, please state the lemma precisely and explain how [24, Theorem 2.2] identifies \\alpha_f(x) with the relevant Lyapunov exponent so that the hypotheses of [24, Lemma 2.4] are satisfied for the point x in question.","section":"Proof of Theorem 1.5, equation (2.12)"}],"recommendation":"major_revision","confidential_remarks":"The morphism-case reduction error is serious because it affects a full branch of the main theorem. If the author can repair the reduction or restrict the theorem statement, the paper would be a solid contribution. The reliance on [24] is acceptable but should be made more transparent. No issues with citation practice or scope have been identified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem is genuinely new and the proof is detailed, but as written the morphism case for arbitrary Y is not proven. At the start of the proof of Theorem 1.5, the reduction to pure-dimensional, regularly embedded Y uses the inequality h_Y ≤ h_{D_1∩...∩D_{N-l}} + O(1). That inequality has the direction backwards: for Y ⊂ Z, standard properties of heights give h_Y ≥ h_Z + O(1), not ≤. The stress-test example is on point: Y=(0:0:1) in P^2, Z={x=0}, and P_n=(2^n:2^n:1) gives h_Y(P_n)=n log 2 and h_Z(P_n)=0. So the reduction collapses, and the theorem's morphism case for arbitrary proper closed subschemes is not established.\n\nWhat the paper does well: the rational-map case, where Y is pure dimensional, regularly embedded, and contained in X_back_f, is the heart of the paper and appears to be worked out carefully. The new technical apparatus—uniform upper bounds on τ(f^{-n}(Y)∩H_1...H_i) via Segre classes and complete intersections—is a real contribution. The author is honest about limitations: the counterexample in Example 3.1 when Y ⊄ X_back_f, the note that the method requires d_1>1, and the explicit reliance on the published [24, Lemma 2.4]. That reliance is not circular and is a minor concern.\n\nThe morphism-case gap is load-bearing. Fixing it might require restricting the statement to pure-dimensional regular Y even when f is a morphism, or finding a different reduction—for example, using divisors vanishing to high order on Y. The asserted inequality is not a typo; it is the step that eliminates the technical hypotheses. Other small issues, like the uncomputed d_2=7 in Example 3.2, are not load-bearing.\n\nWho should read it: someone working on heights of subschemes in arithmetic dynamics will find the rational-map case valuable and the new method worth knowing. The manuscript needs revision before acceptance, but the core ideas are sound and deserve a serious referee. My recommendation: send it to peer review, and ask the referee to focus on the morphism-case reduction. If the author can fix or restrict the statement, this is a solid paper.","headline":"New and substantial result for rational maps, but the morphism-case reduction for arbitrary Y is flawed because the height inequality h_Y ≤ h_Z is backwards.","tokens_in":748,"tokens_out":1142,"would_cite":true,"duration_ms":117178,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37P15","37P55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that along a generic orbit of a dominant rational self-map of a smooth projective variety, the height measuring generalized greatest common divisors among orbit points is asymptotically negligible compared with any ample…","keywords":["Arithmetic dynamics","Generalized greatest common divisors","Height associated with subschemes","Arithmetic degree","Dynamical degree","Rational self-maps","Projective varieties"],"falsifier":"Find a dominant rational map $f$, a closed subscheme $Y$ satisfying the theorem's geometric hypotheses, and a point $x$ with generic orbit and $d_{N-l}(f)^{1/(N-l)} < \\alpha_f(x)$ for which the limsup of $h_Y(f^n(x))/h_H(f^n(x))$ is positive; all quantities are explicit for monomial maps on $\\mathbb{G}_m^N$, where one can compute both heights and check whether the ratio tends to zero.","tokens_in":19731,"feed_emoji":"➗","tokens_out":14081,"duration_ms":135303,"temperature":0.7,"pith_summary":"This paper proves, without assuming the standard Diophantine conjecture used in earlier work, that the height attached to a closed subscheme of codimension $c$ — the 'generalized greatest common divisor' of the orbit points — is asymptotically negligible along a generic orbit of a dominant rational self-map. Concretely, Theorem 1.5 shows $\\lim_{n\\to\\infty} h_Y(f^n(x))/h_H(f^n(x)) = 0$ whenever the orbit of $x$ is generic, the subscheme $Y$ is suitably transverse (or $f$ is a morphism), and the $c$-th dynamical degree satisfies $d_c(f)^{1/c} < \\alpha_f(x)$, the arithmetic degree of $x$. Earlier results on this problem were conditional on the standard Diophantine conjecture; the present proof uses the expansion of the dynamics itself and therefore applies only when the dynamical degrees are not all equal. The conclusion is the expected one: the number of digits of common divisors among orbit coordinates is smaller than the number of digits of the coordinates themselves.","feed_headline":"Generic orbits keep generalized gcd heights negligible","feed_subtitle":"Unconditional proof: generalized gcd heights along generic orbits vanish compared with any ample height.","key_machinery":"One mechanism carries the proof: converting a nonzero section of the ideal-sheaf power $I^r(mH)$ into an effective divisor $D \\sim mH$ that contains the $r$-th thickening of the subscheme, which yields $r h_Z \\leq m h_H + O(1)$ away from $D$. The work is to show such sections exist with $m/r$ as small as $(d_{N-l}(f)^{1/(N-l)} + \\varepsilon)^n$ after replacing $Z$ by $f^{-n}(Y)$. This is done in Proposition 2.10 by bounding the Hilbert function of the thickened $f^{-n}(Y)$ via general hyperplane sections: because $Y$ sits inside the finite-iteration locus and is regularly embedded, the Segre class of $f^{-n}(Y)$ is controlled by the normal bundle, giving $\\tau(f^{-n}(Y) \\cap V, V) \\leq C (f^n)^* H^{N-i} \\cdot H^i$ for general hyperplanes. The resulting bound $h_{f^{-n}(Y)} \\leq (d_{N-l}(f)^{1/(N-l)} + \\varepsilon)^n h_H + O(1)$ outside a proper closed subset is then matched against a lower bound on $h_H$ along the orbit whose exponential rate is arbitrarily close to $\\alpha_f(x)$, forcing the ratio to zero.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1.5. For a smooth projective variety $X$ over $\\overline{\\mathbb{Q}}$ of dimension $N$, a dominant rational self-map $f$, and a proper closed subscheme $Y$ of dimension $l$ with $c = N - l$, the theorem says that if the orbit of $x$ is well-defined and generic, and if $d_c(f)^{1/c} < \\alpha_f(x)$, then $h_Y(f^n(x))/h_H(f^n(x))$ tends to $0$ for every ample height $h_H$. When $f$ is not a morphism, $Y$ must be pure dimensional, a regular embedding in $X$, and contained in the finite-iteration locus $X_f^{\\mathrm{back}}$; for morphisms the theorem shows these extra conditions can be arranged by replacing $Y$ with a general complete intersection containing it. The paper also exhibits a counterexample showing that dropping the finite-iteration condition breaks the conclusion, with the ratio tending to $1$ instead of $0$.","pith_inferences":["The proof reveals a controlling principle the paper does not state as such: the $c$-th dynamical degree, not the first, governs how much 'gcd information' a codimension-$c$ subscheme can accumulate along an orbit; this suggests that for maps with $d_c^{1/c}=d_1$ (such as polarized endomorphisms), a different invariant—likely ramification along $Y$—would be needed to get any vanishing at all.","Because the main estimate is an upper bound with explicit exponential rate, a quantitative version giving a decay rate for $h_Y(f^n(x))/h_H(f^n(x))$ should follow from the same machinery; the paper does not write one down.","The morphism-case reduction, which passes from an arbitrary $Y$ to a general complete intersection containing it, suggests a route to removing the regular-embedding hypothesis for rational maps as well, though the finite-iteration condition would still need to be handled.","The examples give a ready-made computational benchmark: for the monomial map examples one can compute both heights explicitly and verify the ratio's decay, which would serve as a numerical check of the theorem's scope."],"forward_implications":["For morphisms the theorem applies to every proper closed subscheme $Y$: after replacing $Y$ by a general complete intersection containing it, the ratio $h_Y(f^n(x))/h_H(f^n(x))$ tends to $0$ for every generic orbit with $d_{N-l}(f)^{1/(N-l)} < \\alpha_f(x)$.","If the standard conjectures hold that Zariski-dense orbits are generic and that their arithmetic degree equals the first dynamical degree, the hypotheses become simply: the orbit is Zariski dense and $d_{N-l}(f)^{1/(N-l)} < d_1(f)$.","The finite-iteration hypothesis on $Y$ is essential: for $f(x:y:z)=(x^2y:y^3:z^3)$ on $\\mathbb{P}^2$ and $Y=\\{0:0:1\\}$, a point whose orbit is generic and satisfies the degree inequality has $h_Y(f^n(x))/h_H(f^n(x)) \\to 1$.","For monomial maps on the multiplicative torus induced by an integer matrix with $|\\lambda_1| > |\\lambda_N|$, any Zariski-dense orbit with a zero-dimensional $Y$ satisfies the ratio limit, since $d_N(f)^{1/N} = |\\lambda_1\\cdots\\lambda_N|^{1/N} < |\\lambda_1| = d_1(f)$.","For maps with $d_1(f) > d_2(f)$, a point with arithmetic degree equal to $d_1(f)$ automatically has Zariski-dense orbit; combined with the standard genericity conjecture for such orbits, the main theorem applies."],"supporting_citations":[{"why":"Supplies the definition of the finite-iteration locus $X_f^{\\mathrm{back}}$, the Lyapunov-exponent description of arithmetic degree, and Lemma 2.4, the lower height bound along orbits used in the final step.","marker":"[24]"},{"why":"Defines global height functions $h_Y$ associated with closed subschemes, the objects whose ratio to an ample height the theorem shows tends to zero.","marker":"[28]"},{"why":"Contributes the method of bounding $h_Z$ by finding nonzero sections of $I^r(mH)$, which produce a divisor $D \\in |mH|$ containing the $r$-th thickening of $Z$.","marker":"[9]"},{"why":"Provides the intersection-theoretic tools (Segre classes, Chern class projections, projection formula) used to estimate the Hilbert function of thickenings of $f^{-n}(Y)$.","marker":"[8]"},{"why":"Supplies the general-position results for intersections with general hyperplane sections that let the argument cut $Y$ down to lower dimension while preserving regularity.","marker":"[7]"},{"why":"Gives the asymptotic growth of spaces of global sections of $L^m$ and the finiteness of $\\tau$ used in the cohomological criterion of Proposition 2.6.","marker":"[15]"},{"why":"Provides the recursive inequalities that imply log concavity of dynamical degrees, which is used to compare all $d_{N-i}(f)$ with $d_{N-l}(f)^{1/(N-l)}$ in Proposition 2.12.","marker":"[33]"}],"fun_headline_variants":["GCD heights vanish along generic orbits","Generic orbits tame generalized gcd heights","Height ratio collapses under orbit genericity","Dynamical degree bound makes gcd height negligible","Orbits make generalized gcd heights vanish"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The final step leans on a previously established lower bound asserting that along the orbit the height grows at least like $(\\eta \\alpha_f(x))^{mk}$ with $\\eta$ arbitrarily close to $1$; if that bound fails for a point that otherwise meets Theorem 1.5's hypotheses, the proof no longer goes through.","fun_headline_variants_meta":{"raw":{"variants":["GCD heights vanish along generic orbits","Generic orbits tame generalized gcd heights","Height ratio collapses under orbit genericity","Dynamical degree bound makes gcd height negligible","Orbits make generalized gcd heights vanish"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000192,"raw_usage":{"total_tokens":1366,"prompt_tokens":986,"completion_tokens":380,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":602,"completion_tokens_details":{"reasoning_tokens":319}},"tokens_in":602,"tokens_out":380,"duration_ms":5376,"temperature":1.0,"reasoning_tokens":319,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:34:41.452427+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a dominant rational map $f$, a closed subscheme $Y$ satisfying the theorem's geometric hypotheses, and a point $x$ with generic orbit and $d_{N-l}(f)^{1/(N-l)} < \\alpha_f(x)$ for which the limsup of $h_Y(f^n(x))/h_H(f^n(x))$ is positive; all quantities are explicit for monomial maps on $\\mathbb{G}_m^N$, where one can compute both heights and check whether the ratio tends to zero.","supporting_citations":[{"cited_title":"Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal) (2025)","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of the finite-iteration locus $X_f^{\\mathrm{back}}$, the Lyapunov-exponent description of arithmetic degree, and Lemma 2.4, the lower height bound along orbits used in the final step."},{"cited_title":"H.Arithmetic distance functions and height functions in Diophantine geometry","cited_arxiv_id":null,"evidence_quote":"Defines global height functions $h_Y$ associated with closed subschemes, the objects whose ratio to an ample height the theorem shows tends to zero."},{"cited_title":"Bulletin of the London Mathematical Society 56 , 6 (2024), 1939–1950","cited_arxiv_id":null,"evidence_quote":"Contributes the method of bounding $h_Z$ by finding nonzero sections of $I^r(mH)$, which produce a divisor $D \\in |mH|$ containing the $r$-th thickening of $Z$."},{"cited_title":"2 of Ergebnisse der Mathematik und ihrer Gren- zgebiete","cited_arxiv_id":null,"evidence_quote":"Provides the intersection-theoretic tools (Segre classes, Chern class projections, projection formula) used to estimate the Hilbert function of thickenings of $f^{-n}(Y)$."},{"cited_title":"Springer Monographs in Mathematics","cited_arxiv_id":null,"evidence_quote":"Supplies the general-position results for intersections with general hyperplane sections that let the argument cut $Y$ down to lower dimension while preserving regularity."},{"cited_title":"Positivity in algebraic geometry","cited_arxiv_id":null,"evidence_quote":"Gives the asymptotic growth of spaces of global sections of $L^m$ and the finiteness of $\\tau$ used in the cohomological criterion of Proposition 2.6."}],"review_version":1}