Moduli Spaces of Polynomial Maps and Multipliers at Small Cycles

The paper’s Main Theorem states that for any degree d≥2, the multiplier map using periods 1 and 2, Mult(2)_d, is finite and birational onto its image, and for d=2,3 the fixed-point multiplier map Mult(1)_d is an isomorphism onto its image. This is explicitly stated and proved by combining (i) a degeneration-detection result (Theorem A) implying properness (hence finiteness over an affine target) and (ii) a generic injectivity statement (Theorem C) via a local Jacobian analysis around z↦z^d (with the μ_{d−1}-action), together yielding the Main Theorem. The same structure appears in the candidate solution: (A) boundedness of period-1 and 2 multipliers implies properness; (B) generic injectivity; hence finite and birational. For d=2,3 the candidate’s explicit computations agree with Example 17 and Example 18 in the paper (e.g., s1=2 and s2=4c for d=2; and for d=3, s1=6−3a, s2=9−6a, s3=4a^3−12a^2+9a+27 b^2, recovering a and b^2). The paper also records the fixed-point-map degree (d−2)! and the non-quasi-finiteness/surjectivity issues for d≥4, in line with the candidate’s remarks. Minor differences are methodological (the paper deduces finiteness from properness of the holomorphic map into an affine space; the candidate cites the standard algebraic criterion “proper + quasi-finite ⇒ finite”), but the logical flow and conclusions match. Therefore, both are correct and essentially follow the same proof strategy. Key confirmations: Main Theorem and setup (period-1 and 2 multipliers) as in the abstract and introduction, the properness-to-finiteness step, generic injectivity (Theorem C), the explicit d=2 and d=3 formulas, and the degree of Mult(1)_d. Citations: abstract and Main Theorem ; construction of multiplier morphisms and dynatomic/multiplier polynomials ; moduli-slice and μ_{d−1}-action ; properness and finiteness of Mult(2)_d via Theorem A and Corollary 35/Lemma 36/Corollary 37 ; generic injectivity (Theorem C) and its setup (Lemma 71) ; fixed-point map degree and limitations for d≥4 ; explicit d=2 and d=3 computations (Examples 17–18) .