{"id":"e75eedb8-f377-46a7-9121-32bc8d725ff5","arxiv_id":"2411.13522","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Provides an explicit power-saving asymptotic for counting points by pullback height under morphisms between projective spaces over number fields.","lead":"A new theorem counts K-rational points P with H(f(P))≤X for any morphism f between projective spaces over a number field, giving an explicit main term and a power-saving error term. The paper also promises a canonical-height counting formula, but that part contains an internal inconsistency.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.11's canonical-height constant omits the norm of the coefficient ideal of iterated lifts; the proof's redistribution of H is algebraically false.","rationale":"The reader's verdict of CONDITIONAL is appropriate: the main counting theorem for Weil heights (Theorem 1.3) is not directly damaged, but the canonical-height formula in Proposition 2.11 is presented as a central consequence and is currently not established. The reader identified a discrepancy between the displayed formula and the 'In fact' local-factor limits; my closer analysis shows the specific algebraic error: the proof redistributes H(f^i∘g) by dropping Nm⟨F^i∘G⟩. This is verifiable and gives a concrete reason the formula can fail. The reader's identified weakest assumption about Widmer's lemma is a secondary technical concern; it does not need to be resolved for the canonical-height issue to demand a correction. I therefore keep the verdict CONDITIONAL, requiring a corrected Proposition 2.11 and a recomputation of the affected constants and examples before the canonical-height claims can be accepted.","tokens_in":58519,"tokens_out":25923,"duration_ms":249728,"concrete_test":"For f=(2X^2+Y^2 : X^2+2Y^2) over Q, compute c_Q(f^i) via Theorem 1.3 for i=1,2,3, numerically or symbolically, and compare the sequence with the first displayed formula in Proposition 2.11. If the limit differs from the formula by the factor lim Nm⟨F^i⟩^{(m+1)/d^i}, the proposition is false. Alternatively, verify the algebraic identity in the proof: for i=2, H(F^2)^{2/4} equals ∏_v |F^2|_v^{2/4} only after multiplying by Nm⟨F^2⟩^{1/2}=√3, exposing the missing factor.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Proposition 2.11 asserts\nH(f^i∘g)^{n(m+1)/d^i e} = ∏_v |F^i∘G|_v^{n_v(m+1)/d^i e}.\nThis equality ignores the factor Nm⟨F^i∘G⟩^{(m+1)/d^i e} in the definition of the absolute Weil height. Unless this norm factor tends to 1 as i→∞, the displayed formula for \\hat c_{K,f}(g) is missing a nontrivial constant. The omission is not hypothetical: for f=(2X^2+Y^2 : X^2+2Y^2) on P^1 over Q, the lift F^2 has coefficients 9, 12, 6 in the first coordinate and 6, 12, 9 in the second, so Nm⟨F^2⟩=3; the missing factor is 3^{(m+1)/4}, and the limit of such factors is not generally 1. The 'In fact' local-factor formula is the limit of the unnormalized c_{K,v}(f^i∘g) and differs from the advertised global constant by exactly the height-normalization factors, so it does not repair the displayed formula. The main Weil-height theorem (Theorem 1.3) appears unaffected, but the canonical-height counting formula advertised in the abstract and used in the dynamical Schanuel theorem is unsupported as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies, for a nonconstant morphism f:P^m→P^M defined over a number field K, the counting function N_{f*H,P^m(K)}(X)=#{P∈P^m(K):H(f(P))≤X}. Theorem 1.3 asserts an asymptotic with an explicit main term c_K(f)X^{n(m+1)/d} and a power-saving error term whose constants depend on f and K through explicit invariants, following Schanuel's strategy with a new 'excess divisor' machinery. The paper then derives Theorem 2.1, counting points in f(P^m(K)) with respect to H, and Theorem 2.7, a 'dynamical Schanuel' asymptotic for the Call-Silverman canonical height; Proposition 2.11 gives an explicit formula for the limit constant of that asymptotic.","tokens_in":58750,"tokens_out":24149,"duration_ms":232506,"significance":"If Theorem 1.3 is correct, it is a substantial quantitative contribution to the counting of rational points with respect to pullback heights, going beyond previously known Tamagawa-measure asymptotics by providing explicit error terms and explicit dependence on the morphism. The excess-divisor construction, the resultant ideal, and the local-density formalism are potentially useful new tools. Theorem 2.1's thin-set application is a natural and valuable consequence. The paper is carefully structured and contains detailed proofs of the main counting theorem. However, the explicit canonical-height formula in Proposition 2.11 is not supported by the given proof, and the advertised 'formula' for the canonical-height counting function must be corrected or substantially weakened.","major_comments":[{"comment":"The proof of Lemma 9.15 asserts, after equation (83), that Widmer's hypothesis c_v≤1 is superfluous, with the justification 'inspecting his proof ... reveals that this requirement is superfluous'. Since Theorem 1.3's power-saving error term depends directly on applying [39, Lemma 7.1] with possibly c_v>1, this is a load-bearing step and should be justified explicitly, either by reproducing the argument or by quoting a version of Widmer's lemma that allows c_v>1. If the assertion is false, the stated error bound in Theorem 1.3 does not follow as written for arbitrary lifts of f.","section":"Section 9.4, Lemma 9.15"}],"minor_comments":[{"comment":"The notation |\\hat F∘G|_v := lim |F^i∘G|_v^{1/d^i} should be introduced before the 'In fact' paragraph, and the relation between this quantity and the archimedean threshold in the first display should be stated explicitly, since the two displays currently appear contradictory.","section":"Section 2.2, Proposition 2.11"},{"comment":"The function J_{OK,s}(R) is used before it is defined; it should be defined explicitly, for instance as the Jordan totient analogue J_{K,s}(R)=Nm R^s∏_{p|R}(1-Nm p^{-s}).","section":"Section 7, Lemma 7.5"},{"comment":"In part (iii), the equality condition 'δ(i)=0 for all i not divisible by d' should be stated together with the observation that this condition is not vacuous, as Example 6.5 demonstrates; the current wording could be misread as a definition.","section":"Section 6, Corollary 6.4"},{"comment":"In the formula for G_{S,2}(x,y), the Iverson bracket [|x|_2=|y|_2] is used without definition; a brief explanation of the notation and of why the formula holds would improve readability.","section":"Section 2, Example 2.14"}],"recommendation":"major_revision","confidential_remarks":"The main counting theorem, Theorem 1.3, appears to be a solid and valuable contribution if the technical application of Widmer's lemma can be justified. The serious problem is Proposition 2.11: the proof omits the norm factor in the height, so the explicit canonical-height constant is incorrect as stated. The authors should either correct the constant by tracking the limiting norm factor and the full normalization, or substantially weaken the claim to an existence statement and remove or qualify the numerical examples. I would not recommend rejection, because the central Schanuel-type result is likely salvageable, but the advertised canonical-height formula cannot stand in its present form. The unproved assertion about Widmer's c_v≤1 should also be addressed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: Theorem 1.3 is the real thing. It gives an explicit power-saving error term for N_{f*H}(X) with constants that depend on f through a resultant ideal, Lipschitz data, and local volumes. Nothing in Franke–Manin–Tschinkel or Siu has this level of explicitness for M ≥ m. The excess-divisor machinery in Section 5 is a genuine new tool, and the proof follows Schanuel's strategy with several technical gaps filled. I found the main theorem convincing on close reading; the counting, Möbius inversion, and geometry-of-numbers steps line up.\n\nThe advertised canonical-height application also survives scrutiny, though I had to work to see it. The reader's report and the stress-test both say the proof of Proposition 2.11 omits Nm<F^i∘G>. That is not right. The product in H(f^i∘g)^n = ∏_v |F^i∘G|_v^{n_v} runs over all places, including the finite ones; the finite product is exactly 1/Nm<F^i∘G>. So the norm is built into the product, not omitted. Likewise, the \"In fact\" local factors with |\\hat F∘G|_v are the unnormalized local factors, while the displayed global formula is the normalized limit; they differ by precisely the height denominator. No contradiction. The conditional verdict from the reader was based on a misreading.\n\nSoft spots: the paper is long, and the exposition in Section 2.2 could be clearer about which local factors have been normalized. Example 2.13's archimedean constant for T_d relies on a \"WolframAlpha\" evaluation and a Stirling sketch, which is informal but not load-bearing. The claim that Widmer's c_v ≤ 1 condition is superfluous is plausible—if c_v > 1 one can just replace it by 1—though the paper does not reproduce that part of the proof. None of this touches the main theorem.\n\nBottom line: anyone counting rational points in images of morphisms or working with canonical heights will want this. The paper deserves a serious referee; I would send it out and let the referee check Lemma 9.15 and the canonical-height section, but I would not desk reject it.","headline":"Main theorem is new and sound; the reported flaw in Proposition 2.11 does not survive reading the product over all places.","tokens_in":59325,"tokens_out":11811,"would_cite":true,"duration_ms":117683,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G50","14G05","37P30"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any nonconstant morphism between projective spaces over a number field K, the number of K-rational points with H(f(P)) ≤ X is c_K(f) X^{n(m+1)/d} plus an explicit power-saving error term whose constants are expressed through geometric…","keywords":["heights","number fields","morphisms of projective spaces","counting rational points","explicit error term","resultant ideal","canonical height","excess divisor"],"falsifier":"Take the powering map $(x_0:\\dots:x_m) \\mapsto (x_0^d:\\dots:x_m^d)$ over $\\mathbb{Q}$, for which the formula is explicit ($c_{K,v}(f)$ known, $d=1$ recovers the classical count), and compute the exact difference $N_{f^*H,P^m(\\mathbb{Q})}(X) - c_{\\mathbb{Q}}(f)X^{n(m+1)/d}$ for a range of $X$; if the difference does not decay like $X^{(n(m+1)-1)/d}$ (up to log $X$), the error bound in Theorem 1.3 is wrong. For a sharper test, compute with a quadratic map over $\\mathbb{Q}$ of the form given in Example 6.5, where the nonarchimedean local densities are known exactly, and check the predicted constant $c_{K,0}(f)$ against the formula $\\sum_l \\operatorname{Nm}(l)^{(m+1)/d} \\delta_f(l)$.","tokens_in":58240,"feed_emoji":"🧮","tokens_out":12814,"duration_ms":121648,"temperature":0.7,"pith_summary":"The paper proves a counting formula with an explicit error term for the number of K-rational points P in projective m-space whose pullback height H(f(P)) under a morphism f of degree d to projective M-space over a number field K is at most X. The main term is $c_K(f) X^{n(m+1)/d}$, and the error term has order $X^{(n(m+1)-1)/d}$ up to logarithmic factors, with the constants depending on f and K through explicitly defined geometric data: the height of f, a bound on the resultant ideal, Lipschitz constants for the boundary of the local archimedean fundamental domains, and nonarchimedean local volumes. This is the pullback-height analogue of the classical asymptotic counting formula for points of bounded height in projective space, and it comes with a uniformly stated error term. The paper then derives counting formulae for the image f(P^m(K)) and for points of bounded canonical height in arithmetic dynamics.","feed_headline":"Explicit error terms govern height point counts under morphisms","feed_subtitle":"This gives a power-saving asymptotic with explicit constants for points whose pullback height is bounded by X.","key_machinery":"The load-bearing object is the excess divisor $\\ell_f(x) = \\langle F(x)\\rangle / (\\langle x\\rangle^d \\langle F\\rangle)$, a fractional ideal that records the extra valuations picked up by $F(x)$ beyond the generic $\\langle x\\rangle^d \\langle F\\rangle$. The paper shows that $\\ell_f$ is everywhere integral, divides the resultant ideal of f, and is periodic: it factors through the reduction map from $K^{m+1}\\setminus\\{0\\}$ to the finite projective space $P^m(O_K/\\operatorname{Res} f)$. Fibering the counting problem over ideal classes and over the finitely many possible excess divisors, the count of points with prescribed $\\ell_f$ becomes a union of cosets of a lattice, and a generalized Chinese remainder theorem plus an explicit lattice-point counting principle with a Lipschitz boundary estimate for the expanding domain $D_{F,K}(T)$ yields the main term and the power-saving error. The nonarchimedean local constants $c_{K,v}(f)$ are computed as weighted sums of the local densities $\\delta_{f,v}(i)$, and are strictly larger than the v-adic volume of $D_{f,v}$ in general.","core_discovery":"The central discovery is that the counting function for the pullback height satisfies $N_{f^*H,P^m(K)}(X) = c_K(f) X^{n(m+1)/d} + E(X)$, with a fully explicit error bound of order $X^{(n(m+1)-1)/d}(1 + \\log_+ X^{1/d})$ whose implicit constant depends on m and K and on the morphism through the finite list of invariants: $N_f$ and $L_f$, describing how the boundary of each archimedean fundamental domain $D_{f,v} = \\{|F(z)|_v \\le |F|_v\\}$ can be covered by Lipschitz images of the unit cube; the constants $C^0_f$ and $C^\\infty_f$ coming from the local comparison inequalities between $|F(z)|_v$ and $|z|_v^d$; the nonarchimedean local constants $c_{K,0}(f)$; and the height $H(f)$. The main constant factors as $c_K(f) = c_K(m) c_{K,\\infty}(f) c_{K,0}(f)/H(f)^{n(m+1)/d}$, where $c_K(m)$ is the classical constant counting points of bounded height in projective space. The proof achieves this by fibering the count over ideal classes and over finitely many 'excess divisors' $\\ell_f(x) = \\langle F(x)\\rangle / (\\langle x\\rangle^d \\langle F\\rangle)$, proving these divisors are bounded by the resultant ideal, are periodic modulo reduction to a finite projective ring, and have well-defined local densities, and then applying an explicit lattice-point counting principle to the resulting homogeneously expanding domains.","pith_inferences":["The finiteness and periodicity of the excess divisor map means the constants $c_K(f)$ are computable in principle from finite data: residues modulo the resultant ideal together with the volume of the archimedean fundamental domains, so one could implement the formula for explicit morphisms and compare against direct point counts.","The method appears to transfer to counting points of bounded height in images of morphisms between more general varieties equipped with an equivariant height, since the only geometric input is the finite periodic structure of the excess divisors and the Lipschitz class of the archimedean level sets.","In the dynamical setting, the limit constant $\\hat{c}_{K,f}(g)$ is a measure of how the canonical height's unit ball differs from the Weil height's unit ball; the paper's examples show this ratio is not 1 in general, and one could test numerically whether $\\hat{c}_{K,f}(\\mathrm{id})$ converges to the naive constant as the degree of f grows.","The strict inequality between the nonarchimedean local factor and the v-adic volume suggests that 'volume' of the pullback of the unit ball under a morphism is not the right invariant unless the excess valuations are multiples of the degree, which may be relevant when defining Tamagawa measures for pullback line bundles in more general height-counting conjectures."],"forward_implications":["The number of K-rational points in the image $f(P^m(K))$ with Weil height at most $X$ is asymptotic to $(c_K(f)/\\gamma) X^{n(m+1)/d}$ with power-saving error, where $\\gamma$ is the number of K-rational mapping symmetries of f; the error has the same shape as in Theorem 1.3 up to the thin-set error coming from points with non-generic fibre size.","For an endomorphism f of degree $d \\ge 2$, the number of points whose canonical height satisfies $\\hat{h}_f(P) \\le X$ is asymptotic to a constant times $X^{n(m+1)}$, where the constant is a limit of the constants $c_K(f^i \\circ g)$ and admits an explicit formula in terms of the dynamical Green's functions of f.","The canonical-height constant is invariant under iterating f and under conjugation by automorphisms defined over K, but changes under conjugation defined over extensions of K, so the canonical height genuinely redistributes points compared to the Weil height.","The nonarchimedean local factors $c_{K,v}(f)$ are not the 'obvious' v-adic volumes: strict inequality $c_{K,v}(f) > \\mu_v(D_{f,v})$ holds whenever some excess valuation is not divisible by d, as shown by explicit examples.","For the powering map and for maps with good reduction at all but finitely many places, the constants simplify to known values, recovering the classical projective-space counting formula when $d=1$."],"supporting_citations":[{"why":"Supplies the lattice-point counting theorem and the Lipschitz-class estimate for boundaries of expanding domains used to obtain the explicit error term.","marker":"[39]"},{"why":"Provides the volume formula for the homogeneously expanding domain D_{F,K}(T) and the counting principle for lattice points in such domains.","marker":"[23]"},{"why":"The classical proof strategy for counting points of bounded height in projective space, which this paper extends by fibering over excess divisors.","marker":"[32]"},{"why":"Defines the canonical height attached to a morphism, which is the object of the dynamical counting application.","marker":"[3]"},{"why":"Establishes the asymptotic main term cX^{n(m+1)/d} for pullback heights, which Theorem 1.3 refines to an explicit power-saving error term.","marker":"[10]"},{"why":"Provides the convergence and explicit properties of dynamical Green's functions used to compute the limiting constants for canonical-height counting.","marker":"[34]"}],"fun_headline_variants":["Explicit error terms for counting points under morphisms","Power-saving count of points with bounded pullback height","Explicit asymptotic for height counts under morphisms","Morphism height counts get explicit error bounds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The explicit power-saving error term depends on applying a lattice-point counting lemma whose stated hypothesis (a certain comparison constant at each archimedean place must not exceed 1) the paper declares superfluous without reproducing the proof of that relaxation; if the relaxation fails, the error bound as stated may not hold for arbitrary lifts of f.","fun_headline_variants_meta":{"raw":{"variants":["Explicit error terms for counting points under morphisms","Power-saving count of points with bounded pullback height","Explicit asymptotic for height counts under morphisms","Morphism height counts get explicit error bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000173,"raw_usage":{"total_tokens":1300,"prompt_tokens":990,"completion_tokens":310,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":606,"completion_tokens_details":{"reasoning_tokens":250}},"tokens_in":606,"tokens_out":310,"duration_ms":3863,"temperature":1.0,"reasoning_tokens":250,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:23:45.201571+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the powering map $(x_0:\\dots:x_m) \\mapsto (x_0^d:\\dots:x_m^d)$ over $\\mathbb{Q}$, for which the formula is explicit ($c_{K,v}(f)$ known, $d=1$ recovers the classical count), and compute the exact difference $N_{f^*H,P^m(\\mathbb{Q})}(X) - c_{\\mathbb{Q}}(f)X^{n(m+1)/d}$ for a range of $X$; if the difference does not decay like $X^{(n(m+1)-1)/d}$ (up to log $X$), the error bound in Theorem 1.3 is wrong. For a sharper test, compute with a quadratic map over $\\mathbb{Q}$ of the form given in Example 6.5, where the nonarchimedean local densities are known exactly, and check the predicted constant $c_{K,0}(f)$ against the formula $\\sum_l \\operatorname{Nm}(l)^{(m+1)/d} \\delta_f(l)$.","supporting_citations":[{"cited_title":"Counting primitive points of bounded height","cited_arxiv_id":null,"evidence_quote":"Supplies the lattice-point counting theorem and the Lipschitz-class estimate for boundaries of expanding domains used to obtain the explicit error term."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the volume formula for the homogeneously expanding domain D_{F,K}(T) and the counting principle for lattice points in such domains."},{"cited_title":"Heights in number fields","cited_arxiv_id":null,"evidence_quote":"The classical proof strategy for counting points of bounded height in projective space, which this paper extends by fibering over excess divisors."},{"cited_title":"Call and Joseph H","cited_arxiv_id":null,"evidence_quote":"Defines the canonical height attached to a morphism, which is the object of the dynamical counting application."},{"cited_title":"Manin, and Yuri Tschinkel","cited_arxiv_id":null,"evidence_quote":"Establishes the asymptotic main term cX^{n(m+1)/d} for pullback heights, which Theorem 1.3 refines to an explicit power-saving error term."},{"cited_title":"Silverman","cited_arxiv_id":null,"evidence_quote":"Provides the convergence and explicit properties of dynamical Green's functions used to compute the limiting constants for canonical-height counting."}],"review_version":1}