A Bezoutian approach to orthogonal decompositions of trace forms or integral trace forms of some classical polynomials. (English) Zbl 1360.11064
Summary: As U.Helmke and P.A.Fuhrmann pointed out [Linear Algebra Appl. 122–124, 1039–1097 (1989; Zbl 0679.93009)], Bezoutian approaches have been considered to be fruitful for the study of trace forms. In this article, we study orthogonal decompositions of trace forms or integral trace forms of some classical polynomials via Bezoutians. In Section 3, we give another proof of a theorem of W. Feit [J. Algebra 104, 231–260 (1986; Zbl 0609.12005)] about orthogonal decompositions of trace forms of generalized Laguerre polynomials. In Section 4, we consider integral trace forms of certain irreducible trinomials and give their orthogonal decompositions explicitly. Then, in Section 5, we obtain their canonical forms over \(\mathbb{Z}_p\) the ring of \(p\)-adic integers.
MSC:
| 11E08 | Quadratic forms over local rings and fields |
| 11E12 | Quadratic forms over global rings and fields |
| 11R04 | Algebraic numbers; rings of algebraic integers |
| 15A63 | Quadratic and bilinear forms, inner products |
| 15B05 | Toeplitz, Cauchy, and related matrices |
| 15B36 | Matrices of integers |
Keywords:
Bezoutian; trace form; integral trace form; generalized Laguerre polynomial; Hermite polynomial; trinomialReferences:
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