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Remainder Determinants and Mixed Confluence

Abstract

Monic division transforms modified moment determinants, and fixed columns preserve the confluence order of the varying columns.

Theorem 1.1 (Monic multiplier determinants).

Lean statement: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinColumnDeterminant.multiplier_remainders

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinColumnDeterminant.multiplier_remainders (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Johann Cigler (2022). Some remarks and conjectures about Hankel determinants of polynomials which are related to Motzkin paths. DOI: 10.48550/arXiv.2204.09910. URL: https://arxiv.org/abs/2204.09910v4.

Commentary.

Let R be a nontrivial commutative ring, let p_i be monic polynomials of degree i, and let ell be an R-linear functional with ell(p_i p_j) equal to one when i equals j and zero otherwise. For any monic polynomial g of degree h and any nonnegative n, the determinant of ell(g y^(i+j)) for indices i and j below n equals (-1)^(nh) times the h by h determinant whose entry in row i and column j is the coefficient of y^j in the remainder of p_{n+i} modulo g. Integral triangular changes of basis and monic division give the identity, including empty matrices.

Theorem 1.2 (Confluence with fixed columns).

Lean statement: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinColumnDeterminant.mixed_confluence

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinColumnDeterminant.mixed_confluence (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Johann Cigler (2022). Some remarks and conjectures about Hankel determinants of polynomials which are related to Motzkin paths. DOI: 10.48550/arXiv.2204.09910. URL: https://arxiv.org/abs/2204.09910v4.

Commentary.

Let R be a commutative ring and m and k be nonnegative integers. Choose formal series f_i(y) for rows i below m + k, scalars c_j for j below m, and fixed entries g_{i,j} for j below k. Form A(y) with first m columns f_i(c_j y) and last k columns g_{i,j}. Form J with first m columns the coefficients of y^j in f_i and the same last k columns. Every coefficient of det A below binom(m,2) vanishes, and its coefficient at binom(m,2) is det Vandermonde(c) times det J. Expansion along the fixed columns reduces the formula to alternant confluence.

References