What does this research mean for the field?

In the quadratic family x^2 + c, specific packets of terminal period 3 satisfy a newly established criterion ensuring their entire Kummer towers are maximal, meaning no hidden multiplicative relations exist among the multipliers. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To analyze the algebraic properties, Galois groups, and associated Kummer towers of cycle multipliers for the quadratic family f(x) = x² + c over the rational numbers.

August 31, 2026Open Access

Multiplier towers of the quadratic family x² + c: irreducibility, Galois groups, and maximal Kummer towers

View Full Paper

Ask AI

Bookmark

Authors

SGSylvain Gefffroy

Discussion

Member takes

Implication

Theoretical analysis reveals maximal Kummer towers and symmetric Galois groups in quadratic multiplier systems, indicating an absence of hidden multiplicative relations.

Key Points

To analyze the algebraic properties, Galois groups, and associated Kummer towers of cycle multipliers for the quadratic family f(x) = x² + c over the rational numbers.
Derived quadratic recursions, norm identities, and exact resultant identities for multiplier polynomials Phi_n and parameter polynomials Q_n.
Explicitly computed degree-6 and degree-12 multiplier polynomials for orbits landing on terminal period 3 with preperiods 2 and 3 using symbolic computation.
Established a general algebraic criterion for 2-primary radical extensions of monic irreducible polynomials with symmetric Galois groups.
Demonstrated that the Galois groups over the rationals for the degree-6 and degree-12 multiplier polynomials are the full symmetric groups S_6 and S_12.
Proved that Kummer towers for both investigated packets achieve maximal degree at every depth s, indicating no hidden multiplicative relations exist among the multipliers.
Showed via resultant identities that multiplier products along any level equal a power of 2 up to sign, precluding linear disjointness between such families.

Cite This Study

Sylvain Gefffroy (2026) studied this question.

synapsesocial.com/papers/6a956cd5f20e493292a74e35 https://doi.org/10.5281/zenodo.22179953

View Full Paper

Ask AI

Bookmark

Implication

Theoretical analysis reveals maximal Kummer towers and symmetric Galois groups in quadratic multiplier systems, indicating an absence of hidden multiplicative relations.

Multiplier towers of the quadratic family x² + c: irreducibility, Galois groups, and maximal Kummer towers

Discussion

Member takes

Implication

Key Points

Cite This Study

Multiplier towers of the quadratic family x² + c: irreducibility, Galois groups, and maximal Kummer towers

Discussion

Member takes

Implication

Key Points

Cite This Study

Also Consider

Also Consider