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To develop a structure theory for the sin²-type algorithm applied to complex cubic fields and address Karpenkov's Problem 4.

July 13, 2026Open Access

A dual reformulation of the complex sin2-algorithm: exact identities, descent, and finiteness

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To develop a structure theory for the sin²-type algorithm applied to complex cubic fields and address Karpenkov's Problem 4.
Introduced deterministic sin²-type algorithm for complex cubic fields, focusing on the complex-signature case.
Utilized a minimization selection rule involving hyperbolic distances and embedded real quantities in exact form.
Proved several theorems regarding growth bounds, state finiteness, and periodicity under specific hypotheses.
Showed that the module can grow at most by the factor φ²=2.618… in one step.
Established a finiteness theorem for states with constrained module and height, providing explicit constants.
Proved height-descent theorem indicating that height cannot exceed certain fixed maximum values.

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Randomized trial reveals exact identities and finiteness in complex cubic fields, indicating significant developments in algorithm structure.

Ludovic Tagnon (2026) studied this question.

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Randomized trial reveals exact identities and finiteness in complex cubic fields, indicating significant developments in algorithm structure.