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Mixed Remainder and Coefficient Coordinates

Abstract

A triangular change of polynomial coordinates separates remainders modulo v from the first d coefficients.

Theorem 1.1 (The coordinate determinant multiplier).

Lean statement: D5/S3/Combinatorics/CatalanPowerHankel/CiglerElevenReduction.coordinate_change

Proof. Machine-checked in Lean as D5/S3/Combinatorics/CatalanPowerHankel/CiglerElevenReduction.coordinate_change (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Johann Cigler (2023). Some experimental observations about Hankel determinants of convolution powers of Catalan numbers. DOI: 10.48550/arXiv.2308.07642. URL: https://arxiv.org/abs/2308.07642v2.

Commentary.

Let k be a nonnegative integer, let d be a positive integer, let v be a monic rational polynomial of degree k, and let Q_i be any k+d rational polynomials. Form a square matrix with entry [X^j](Q_i mod v) for columns j less than k and entry [X^(j-k)]Q_i for the remaining columns. Its determinant is v(0)^d times the determinant of the coefficient matrix with entry [X^j](Q_i mod (X^d v)) for all columns from zero through k+d-1. Remainders mean polynomial remainders on division by a monic polynomial. The monic basis 1,X,…,X^(k-1),v,Xv,…,X^(d-1)v makes the coordinate transformation block triangular with an identity block and a multiplication block of determinant v(0)^d. No nonzero constant coefficient is required.

References