{"id":"5aa8fd8e-1ea8-4fde-a89b-5b6121bf5fcb","arxiv_id":"1908.07361","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For prime d, the Misiurewicz polynomial G_{d,m,n} has no more rational irreducible factors than its reduction modulo d, which yields new irreducibility families and the first bounds for periods above three.","lead":"This paper proves new results on Misiurewicz polynomials, the equations whose roots encode polynomial maps x^d+c whose critical point lands on a repeating cycle. It gives the first provable statements for cycle lengths greater than three, and converts a classical question about Galois conjugates into a simpler check over finite fields.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.6's containment proof has an invalid descent step: a_n^{l+1} in J only yields d in J when l+1 ≤ M, and 'repeating' cannot overcome the opposite inequality.","rationale":"The reader's weakest_assumption identified the imported Theorem 2.1 as the main risk; I agree that the external dependence is a serious concern. However, the most concrete unproved step inside this paper is the ⊆ direction of Proposition 2.6. In good faith, I think the proposition is probably true: the ideals I_i are comaximal because the f_i are distinct irreducible factors modulo a prime, and comparing prime exponents should force (a_n) = ∏ I_i. But the proof as written does not make that argument, and its descent is formally invalid. Since Theorem 1.4, Corollaries 1.5 and 1.6, Theorem 1.9, and Corollary 3.5 all rely on Proposition 2.6 directly or via (2.3), the central claim is not fully certified as written. I do not see an outright counterexample or internal contradiction; the issue is a key lemma with an incomplete proof sketch rather than a demonstrated failure. This supports the reader's CONDITIONAL verdict rather than changing it, so I recommend keeping the verdict unchanged.","tokens_in":11614,"tokens_out":29722,"duration_ms":292890,"concrete_test":"Verify Proposition 2.6 in the first case with k > M: take d = 2, m = 2, n = 5, so M_{2,5} = 2 and G_{2,0,5}(c) mod 2 = (c^5+c^2+1)(c^5+c^3+1)(c^5+c^3+c^2+c+1), giving k = 3 irreducible factors. Pick a root c0 of G_{2,2,5}(c) and compute, in O_{Q(c0)}, the ideal (a_5) and the product (2, \\tilde f_1(c0))(2, \\tilde f_2(c0))(2, \\tilde f_3(c0)), with \\tilde f_i any integer lifts of the three mod-2 factors. If the ideals are unequal, Proposition 2.6 is false and Theorem 1.4 fails. If they are equal, the descent step can be replaced by a comaximality/valuation proof; writing such a proof would close the gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the ⊆ direction of Proposition 2.6, the author writes k = M l + q and groups the ideals (d, \\tilde f_i(c0)) into blocks of size M. For each block he shows a_n lies in the product of the block's ideals, hence a_n^{l+1} lies in J = ∏_{i=1}^k (d, \\tilde f_i(c0)). He then concludes that d ∈ J if l+1 ≤ M, which is correct because (a_n)^M = (d) ⊆ (a_n)^{l+1}. But when l+1 > M, the inclusion runs the other way: (a_n)^{l+1} ⊆ (a_n)^M, so the fact that a large power of a_n lies in J carries no information about d or a_n^M. The 'obvious' terminating descent does not supply d ∈ J; it repeats the same inference with the same obstruction. This gap matters because (2.3) is the explicit factorization of (d) used in the contradiction argument of Theorem 1.4. A repaired proof can likely be obtained by first showing the ideals (d, \\tilde f_i(c0)) are pairwise comaximal, since the f_i are distinct mod d, and then comparing prime-exponent vectors, but that argument is absent from the paper.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Misiurewicz polynomials G_{d,m,n}(c) for the unicritical family f_{c,d}(x)=x^d+c, with d prime. The main result, Theorem 1.4, claims that the number of irreducible factors of G_{d,m,n} over Q is bounded above by the number of irreducible factors of the reduction G_{d,0,n} in F_d[c], so that irreducibility mod d implies irreducibility over Q. The proof strategy is to factor the ideal (a_n) in the number field K=Q(c_0) attached to a root c_0 of G_{d,m,n} as a product of ideals (d,\\tilde f_i(c_0)), using the critical-orbit ideal relation (a_n)^{M_{m,n}}=(d) imported from the author's earlier paper [5]. Theorem 1.9 states that d does not divide the index [O_K:Z[c_0]], and Corollary 3.5 deduces that the critical orbit elements are square-free. A new corollary, G_{3,m,2} irreducible for all m≥2, is also derived. The paper includes an instructive example for d=2,n=4.","tokens_in":11676,"tokens_out":19300,"duration_ms":201307,"significance":"If Theorem 1.4 is correct, it is a substantial reduction: it ties irreducibility of G_{d,m,n} over Q to the mod d factorization of G_{d,0,n} independent of m, going beyond the previously known period-size-at-most-3 cases and giving the new family G_{3,m,2}. The paper is also honest about its main imported input, Theorem 2.1 from [5], and the explicit factorization example for G_{2,0,4} is correct. However, the proof as written contains two load-bearing gaps: the descent argument in Proposition 2.6 is invalid in the case k>M_{m,n}, and the ideal containment (2.4) in the proof of Theorem 1.4 does not follow from the displayed equality. Additional loose ends in the proof of Theorem 1.9 involve the use of Lemma 2.3 for a degree equality and the choice of the polynomial G in (3.1). These issues make the central claims currently unproved, although the overall strategy appears plausible and likely repairable.","major_comments":[{"comment":"The descent step in the proof of the containment (2.1) is invalid. From the block decomposition the author obtains a_n^{l+1} ∈ J = ∏_{i=1}^k (d,\\tilde f_i(c_0)). The conclusion d ∈ J is justified only when l+1 ≤ M_{m,n}, because then (a_n)^{M_{m,n}} = (d) ⊆ (a_n)^{l+1} ⊆ J. When l+1 > M_{m,n} the inclusion reverses: (a_n)^{l+1} ⊆ (a_n)^{M_{m,n}}, so knowing that a large power of a_n lies in J gives no information about whether d, or a_n^{M_{m,n}}, lies in J. The statement 'repeating the same argument' does not remove this obstruction because k is fixed and the same inequality reappears. This gap matters because equation (2.3), which is derived from Proposition 2.6, is used in the proofs of Theorem 1.4 and Corollary 3.5. A repair is likely available by first proving that the ideals (d,\\tilde f_i(c_0)) are pairwise comaximal, since the reductions f_i are distinct irreducibles in F_d[c], and then using that their product equals their intersection; this argument is absent from the manuscript.","section":"§2, Proposition 2.6 (⊆ direction, k>M_{m,n} case)"},{"comment":"The containment (2.4) does not follow from the equality \\tilde f_1(c_0)^{α_1}...\\tilde f_k(c_0)^{α_k} = -dH_1(c_0). That equality shows only that the single product element lies in the ideal (d). The product ideal ∏ (d,\\tilde f_i(c_0))^{α_i} has generators such as \\tilde f_1(c_0)^{α_1}\\tilde f_2(c_0)^{α_2-1}\\tilde f_3(c_0)^{α_3}..., which are not divisible by d, so the claim that 'all the generators of the product ideal are divisible by d' is false. Consequently the conclusion that some (d,\\tilde f_i(c_0)) must be the unit ideal is unsupported. This is a second load-bearing gap in the proof of Theorem 1.4; a corrected argument would need to compare prime-exponent vectors using the factorization of (d) obtained from a repaired Proposition 2.6, and that argument is not present.","section":"§2, proof of Theorem 1.4, equation (2.4)"},{"comment":"Two points in the proof of Theorem 1.9 need attention. First, Lemma 3.3 uses the equality deg(G_{d,m,n}) = M_{m,n} deg(G_{d,0,n}), citing Lemma 2.3; but Lemma 2.3 is a congruence modulo d and by itself does not determine degrees unless the leading coefficient of G_{d,m,n} is shown not to be divisible by d. Second, the polynomial G in (3.1) is determined by the particular lifts A_i chosen, while Lemma 3.4 supplies a special F with the required resultant properties; the proof never shows that the G in (3.1) can be taken to be that F. These are likely fixable by choosing lifts compatibly from the outset, but as written the argument has a gap.","section":"§3, Lemma 3.3 and the proof of Theorem 1.9"}],"minor_comments":[{"comment":"The notation alternates between p and d for the same rational prime, particularly in Proposition 2.6, Lemma 2.5, and Lemma 3.4; standardizing on one symbol would improve readability.","section":"Throughout"},{"comment":"The sentence 'all Misiurewicz points lie in Q' should read 'lie in \\bar{Q}', since the preceding sentences correctly use \\bar{Q}.","section":"§1, first paragraph"},{"comment":"In the displayed norm computation, 'ap+bk' appears to be a typo for 'ap+bα'.","section":"§2, proof of Lemma 2.4"},{"comment":"The displayed expression for the possible factorization of G_{2,m,4} is ambiguous; the exponent M_{m,4} should apply to each irreducible factor of G_{2,0,4} in F_2[c], i.e. (c^2+c+1)^{M_{m,4}} and (c^4+c+1)^{M_{m,4}}, consistent with Lemma 2.3.","section":"§1, Example 1.8"}],"recommendation":"major_revision","confidential_remarks":"The paper is clearly written and the overall idea is appealing, but the proof of the main theorem has two serious technical gaps that are not merely cosmetic. I would not reject outright because the announced result is plausible and the defects may be repairable with additional ideal-theoretic arguments, but the revision needs to supply those arguments carefully. The heavy reliance on Theorem 2.1 of [5] and on Lemma 3 of [2] should also be made explicit in the introduction, since the main results collapse if those inputs fail."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou should know this paper has one genuinely new result and one genuine proof gap. The new thing is Theorem 1.4: for prime degree d, the number of irreducible factors of G_{d,m,n}(c) over Q is bounded by the number of irreducible factors of G_{d,0,n}(c) modulo d. That is a clean, useful reduction, and it recovers the known irreducibility cases plus G_{3,m,2}. Theorem 1.9, the non-divisibility of d from [O_K : Z[c0]], is also new and looks correct. The paper is a serious contribution to Milnor's conjugacy question, and the reader's conditional verdict is about right.\n\nThe soft spot is Proposition 2.6 in the k > M_{m,n} case. The descent step does not work as written. From a_n^{l+1} lying in the product ideal J, you can only conclude d lies in J when l+1 ≤ M_{m,n}. When l+1 > M_{m,n}, the inclusion runs the opposite way: (a_n)^{l+1} is contained in (a_n)^M, so knowing a large power of a_n is in J tells you nothing about d. \"Repeating the same argument\" does not fix this, because the exponent l+1 does not decrease. The theorem may still be true, and the stress-test note's suggested fix (first prove the ideals (d, \\tilde f_i(c0)) are pairwise comaximal, then compare prime-exponent vectors) is plausible, but that argument is absent. Since (2.3) is the explicit factorization of (d) used in Theorem 1.4, this gap is load-bearing, not cosmetic.\n\nMinor issues: the abstract overclaims slightly—\"first to say something provable\" for period size greater than 3 really means a factor-count bound and an index theorem, not irreducibility. The MAGMA computation in Example 1.8 ships no code or data, so it is only anecdotal. The heavy reliance on your own Theorem 2.1 from [5] is fine, since it is separate work, but it means the whole stack stands or falls with an imported result.\n\nBottom line: worth a serious referee. A good referee will spot the Proposition 2.6 gap and send it back for revision. If the gap is fixable, this becomes a solid paper. I would not cite it in its current form, but I would bring it to a reading group to work through the descent.\n\nRecommendation: accept for peer review, expect revision.","headline":"Genuinely new reduction for Misiurewicz polynomials, but the main proof has a real gap in Proposition 2.6 that needs a comaximality or valuation argument.","tokens_in":12477,"tokens_out":4582,"would_cite":false,"duration_ms":46684,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R09","37P15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For prime d, a modulo-d polynomial controls Galois conjugacy of all Misiurewicz points of a given type.","keywords":["iteration","post-critically finite","Misiurewicz point","Misiurewicz polynomial","irreducibility","Galois conjugacy","ideal factorization"],"falsifier":"In the open case $d=2$, $n=4$, factor $G_{2,m,4}(c)$ over $\\mathbb{Q}$ for progressively larger $m$; Theorem 1.4 predicts at most two irreducible factors, so a single $m$ with three or more factors would refute it. Independently, for any root $c_0$ of $G_{d,m,n}(c)$, compute the norm of the periodic critical-orbit element $a_n$ in $K=\\mathbb{Q}(c_0)$ and compare its $M_{m,n}$-th power with $\\pm d^{\\deg G_{d,0,n}}$; a mismatch would falsify the ideal-structure theorem on which every main result depends.","tokens_in":11175,"feed_emoji":"🔁","tokens_out":19377,"duration_ms":164036,"temperature":0.7,"pith_summary":"This paper proves that, when $d$ is a prime, the Misiurewicz polynomial $G_{d,m,n}(c)$ — whose roots are the parameter values $c_0$ for which $x^d+c_0$ has a strictly pre-periodic critical orbit of exact type $(m,n)$ — is governed by a single reduced polynomial $G_{d,0,n}(c)$ modulo $d$. The main theorem says that the number of irreducible factors of $G_{d,m,n}(c)$ over $\\mathbb{Q}$ is at most the number of irreducible factors of $G_{d,0,n}(c)$ in $\\mathbb{F}_d[c]$; in particular, irreducible modulo $d$ implies irreducible over $\\mathbb{Q}$. Since irreducibility of $G_{d,m,n}(c)$ over $\\mathbb{Q}$ is equivalent to all Misiurewicz points of type $(m,n)$ being Galois conjugates, this settles that classical question whenever the reduced polynomial is irreducible modulo $d$. The paper also proves that for any root $c_0$ of $G_{d,m,n}(c)$, the prime $d$ does not divide the index $[O_K:\\mathbb{Z}[c_0]]$ in $K=\\mathbb{Q}(c_0)$, and that the critical orbit of $f_{c_0,d}$ consists of square-free elements in $O_K$. These claims cover all periods $n$, going beyond the period-at-most-3 cases known previously.","feed_headline":"Irreducibility mod d forces Galois conjugacy of Misiurewicz points","feed_subtitle":"If the reduced period polynomial is irreducible mod d, all same-type Misiurewicz points are Galois conjugates.","key_machinery":"The engine is an ideal-structure theorem for the critical orbit of $f_{c,d}(x)=x^d+c$: when $d$ is prime and the map is post-critically finite of exact type $(m,n)$, the orbit element $a_i=f^i_{c,d}(0)$ at any multiple $i$ of the period satisfies $(a_i)^{M_{m,n}}=(d)$, where $M_{m,n}=d^{m-1}(d-1)$ if $n\\nmid m-1$ and $(d^{m-1}-1)(d-1)$ if $n\\mid m-1$. The proof converts this into polynomial information by way of three lemmas: $a_n$ equals $G_{d,0,n}(c_0)$ up to a unit in $O_K$, $G_{d,m,n}(c)$ is congruent modulo $d$ to $G_{d,0,n}(c)^{M_{m,n}}$, and the ideal $(d)$ factors as a product of powers of ideals $(d,\\tilde f_i(c_0))$ indexed by the irreducible factors of the reduction of $G_{d,0,n}$. With that factorization in hand, a resultant computation plus the classical index criterion proves the index non-divisibility theorem. The machinery makes the dependence on $m$ disappear modulo $d$, so an infinite family of irreducibility questions reduces to one fixed polynomial over $\\mathbb{F}_d$.","core_discovery":"The paper's central claim is a factorization-bound theorem: for $d$ prime and any $m \\ge 2$, the number of irreducible factors of the Misiurewicz polynomial $G_{d,m,n}(c)$ over $\\mathbb{Q}$ is no larger than the number of irreducible factors of the reduced polynomial $G_{d,0,n}(c)$ in $\\mathbb{F}_d[c]$. Because irreducibility of $G_{d,m,n}(c)$ over $\\mathbb{Q}$ is equivalent to the Galois conjugacy of all Misiurewicz points of type $(m,n)$, the theorem gives a uniform answer to the conjugacy question in every case where the reduction is irreducible. The same circle of ideas yields Theorem 1.9: for a root $c_0$ of $G_{d,m,n}(c)$, with $K=\\mathbb{Q}(c_0)$, the prime $d$ does not divide $[O_K:\\mathbb{Z}[c_0]]$, so the splitting of $d$ in $K$ is described by the factorization of $G_{d,m,n}(c)$ modulo $d$; Corollary 3.5 then says the critical orbit's non-unit elements are square-free in $O_K$.","pith_inferences":["Beyond the paper, this gives a one-time computational test for the Galois-conjugacy question: for fixed $(d,n)$, factor $G_{d,0,n}(c)$ modulo $d$ once; if it is irreducible, the answer for every preperiod $m \\ge 2$ is uniform, so the infinite family is collapsed to a finite check.","The index theorem opens a route to stability of iterates of $x^d+c$: since non-unit critical-orbit elements are square-free in $O_K$, the standard obstruction of such an element being a $\\pm d$-th power is absent outside the units; checking whether the unit elements can ever be such powers would settle the remaining stability cases.","Example 1.8 turns the $d=2$, $n=4$ case into a searchable problem: any reducible $G_{2,m,4}(c)$ must be a product of two integer polynomials whose reductions modulo $2$ are powers of $c^2+c+1$ and $c^4+c+1$, and a computer search over $m$ can either expose such an $m$ or accumulate evidence for full irreducibility."],"forward_implications":["Whenever $G_{d,0,n}(c)$ is irreducible in $\\mathbb{F}_d[c]$, the polynomial $G_{d,m,n}(c)$ is irreducible over $\\mathbb{Q}$ for every $m \\ge 2$, and all Misiurewicz points of type $(m,n)$ are Galois conjugates; this answers the conjugacy question for those $(d,n)$ once and for all.","The number of irreducible factors of $G_{d,m,n}(c)$ over $\\mathbb{Q}$ is independent of $m$ and bounded by the number of irreducible factors of $G_{d,0,n}(c)$ modulo $d$; for $d=2$, $n=4$, for example, $G_{2,m,4}(c)$ can have at most two factors for every $m$.","For $d=3$ and $n=2$ the theorem produces a new infinite family of irreducible polynomials: since $c^2+1$ is irreducible in $\\mathbb{F}_3[c]$, $G_{3,m,2}(c)$ is irreducible over $\\mathbb{Q}$ for all $m \\ge 2$.","For any root $c_0$ of $G_{d,m,n}(c)$, the splitting of $d$ in $\\mathbb{Q}(c_0)$ is governed by the factorization of $G_{d,m,n}(c)$ modulo $d$, because $d$ does not divide the index $[O_K:\\mathbb{Z}[c_0]]$.","The known irreducibility results for $G_{d,m,1}(c)$, $G_{2,m,2}(c)$, and $G_{2,m,3}(c)$ follow from the same theorem, since their reduced period polynomials are irreducible in $\\mathbb{F}_d[c]$."],"supporting_citations":[{"why":"It supplies Theorem 2.1, the critical-orbit ideal-structure identity $(a_i)^{M_{m,n}}=(d)$, from which Propositions 2.6 and Lemma 3.3 are derived.","marker":"[5]"},{"why":"It supplies Lemma 2.3, the congruence $G_{d,m,n} \\equiv G_{d,0,n}^{M_{m,n}} \\pmod d$, plus the resultant lemma used in Theorem 1.9.","marker":"[3]"},{"why":"It provides the fact that $\\operatorname{Disc}(G_{d,0,n})$ is coprime to $d$, the hypothesis required for Lemma 3.4 and hence Theorem 1.9.","marker":"[2]"},{"why":"It supplies the classical criterion for a prime not to divide the index $[O_K:\\mathbb{Z}[c_0]]$, used to prove Theorem 1.9.","marker":"[4]"},{"why":"It provides the primitive-part lemma used in Lemma 2.2 to identify $G_{d,0,n}(c_0)$ as the primitive divisor of $a_n$.","marker":"[9]"},{"why":"It shows that the Misiurewicz polynomials $G_{d,m,n}(c)$ are integer polynomials, the objects whose factorization is the paper's target.","marker":"[7]"},{"why":"It supplies the rigid divisibility sequence fact used in Lemma 2.2 and the stability criterion that motivates Corollary 3.5.","marker":"[6]"}],"fun_headline_variants":["Mod d irreducibility ties Misiurewicz points to Galois","Prime d unlocks Misiurewicz conjugacy past period 3","Factorization bound cracks Misiurewicz conjugacy","Prime d: Misiurewicz conjugacy for periods >3"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the imported theorem that for a prime $d$ and a post-critically finite map $x^d+c$ of exact type $(m,n)$, the critical-orbit element at the period satisfies $(a_i)^{M_{m,n}}=(d)$; if that identity fails, or needs extra hypotheses, Theorems 1.4 and 1.9 collapse.","fun_headline_variants_meta":{"raw":{"variants":["Mod d irreducibility ties Misiurewicz points to Galois","Prime d unlocks Misiurewicz conjugacy past period 3","Factorization bound cracks Misiurewicz conjugacy","Prime d: Misiurewicz conjugacy for periods >3"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001139,"raw_usage":{"total_tokens":4755,"prompt_tokens":999,"completion_tokens":3756,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":615,"completion_tokens_details":{"reasoning_tokens":3680}},"tokens_in":615,"tokens_out":3756,"duration_ms":24221,"temperature":1.0,"reasoning_tokens":3680,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:25:54.343134+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the open case $d=2$, $n=4$, factor $G_{2,m,4}(c)$ over $\\mathbb{Q}$ for progressively larger $m$; Theorem 1.4 predicts at most two irreducible factors, so a single $m$ with three or more factors would refute it. Independently, for any root $c_0$ of $G_{d,m,n}(c)$, compute the norm of the periodic critical-orbit element $a_n$ in $K=\\mathbb{Q}(c_0)$ and compare its $M_{m,n}$-th power with $\\pm d^{\\deg G_{d,0,n}}$; a mismatch would falsify the ideal-structure theorem on which every main result depends.","supporting_citations":[{"cited_title":"On the orbit of a post-critically finite polynomial of the form $x^d + c$","cited_arxiv_id":"1806.01208","evidence_quote":"It supplies Theorem 2.1, the critical-orbit ideal-structure identity $(a_i)^{M_{m,n}}=(d)$, from which Propositions 2.6 and Lemma 3.3 are derived."},{"cited_title":"Rational maps with a preperiodic critical point","cited_arxiv_id":"1806.11221","evidence_quote":"It supplies Lemma 2.3, the congruence $G_{d,m,n} \\equiv G_{d,0,n}^{M_{m,n}} \\pmod d$, plus the resultant lemma used in Theorem 1.9."},{"cited_title":"On postcritically finite unicritical polynomial s","cited_arxiv_id":null,"evidence_quote":"It provides the fact that $\\operatorname{Disc}(G_{d,0,n})$ is coprime to $d$, the hypothesis required for Lemma 3.4 and hence Theorem 1.9."},{"cited_title":"Uber den Zusammenhang zwischen der Theorie der Ideale und der Theorie der h¨ oheran Kongruenzen","cited_arxiv_id":null,"evidence_quote":"It supplies the classical criterion for a prime not to divide the index $[O_K:\\mathbb{Z}[c_0]]$, used to prove Theorem 1.9."},{"cited_title":"Dynamical Galois groups of trinomials and Odoni's conjecture","cited_arxiv_id":"1609.03398","evidence_quote":"It provides the primitive-part lemma used in Lemma 2.2 to identify $G_{d,0,n}(c_0)$ as the primitive divisor of $a_n$."},{"cited_title":"Misiurewicz points for polynomial m aps and transversality","cited_arxiv_id":null,"evidence_quote":"It shows that the Misiurewicz polynomials $G_{d,m,n}(c)$ are integer polynomials, the objects whose factorization is the paper's target."},{"cited_title":"The density of primes in or bits of zd + c","cited_arxiv_id":null,"evidence_quote":"It supplies the rigid divisibility sequence fact used in Lemma 2.2 and the stability criterion that motivates Corollary 3.5."}],"review_version":1}