Multivariate Lucas polynomials and ideal classes in quadratic number fields


Abstract: In this work, by using Pauli matrices, we introduce four families of polynomials indexed over the positive integers. These polynomials have rational or imaginary rational coefficients. It turns out that two of these families are closely related to classical Lucas and Fibonacci polynomial sequences and hence to Lucas and Fibonacci numbers. We use one of these families to give a geometric interpretation of the 200-year-old class number problems of Gauß, which is equivalent to the study of narrow ideal classes in real quadratic number fields.

Keywords: Pauli matrices, Fibonacci polynomials, Lucas polynomials, çarks, çark hypersurfaces, indefinite binary quadratic forms, class number problems of Gauß, real quadratic number fields, narrow ideal classes

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