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This research explores two sequences of polynomials that yield identical Salem numbers.

March 15, 2026Open Access

Identical Salem Numbers Arising from Two Families of Palindromic Polynomials

Key Points

This research explores two sequences of polynomials that yield identical Salem numbers.
Two sequences of characteristic polynomials were analyzed.
Palindromic polynomials were generated through symmetrization and factorization.
SageMath was used for numerical verifications of Salem polynomial properties.
Two families of palindromic polynomials were found to produce the same Salem numbers.
Three tables were created: characteristic polynomials, palindromic polynomials, and common Salem numbers.

Abstract

AbstractBased on two sequences of characteristic polynomials and subsequent symmetrization andfactorization, we obtain two distinct families of palindromic polynomials. Surprisingly, thesetwo families produce exactly the same Salem numbers. We present three tables: the first twocontain the characteristic polynomials and the resulting palindromic polynomials for each family(20 entries each), and the third lists the common Salem numbers. All polynomials satisfy thedefining criteria of Salem polynomials, verified numerically with SageMath. The computationsalso benefited from the conversational assistance of DeepSeek.

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Emma Helmdach (Fri,) studied this question.

synapsesocial.com/papers/69b5ff3b83145bc643d1b698 https://doi.org/https://doi.org/10.5281/zenodo.18997287

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