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Toeplitz multiplication and graded factorization of determinant recurrences

Max A. Alekseyev, Dmitry I. Khomovsky

arXiv:2609.27268v2math.RAcs.SC

TL;DR

Finite Toeplitz sections acquire corner corrections when their symbols are multiplied, complicating the recurrence spectrum. This paper links those defects to exterior-degree recurrence sectors through graded multiplication and matrix filtrations. The one-sided defect-order recurrences are generically minimal, with successive pieces matching the graded sectors.

  • Problem

    The paper asks how determinant recurrences change under symbol multiplication, since finite-section products differ from the section of the product by corner defects.

    Source: §1 · p. 2

  • Method

    The paper combines graded recurrence profiles for polynomial cores with finite-section defect filtrations that identify boundary corrections with exterior-degree sectors.

    Source: §1 · p. 2, §9 · p. 30

  • Results

    For every one-sided defect order, the cumulative recurrence is generically minimal, and its successive associated graded pieces are exactly the recurrence sectors.

    Source: §3.2 · p. 12

  • Takeaways & Limitations

    The framework provides a factor-level description of how finite-section boundary effects generate recurrence spectra for structured banded Toeplitz products.

    Source: §9 · p. 30

    At exceptional coefficient specializations, the canonical annihilator may be nonminimal because modes can coincide, vanish from the sequence, or arise in confluent cases.

    Source: §2 · p. 4

Abstract

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We study how determinant recurrences of banded Toeplitz matrices behave when their Laurent symbols are multiplied. Two classical structures underlie the problem. Clean banded Toeplitz determinants have a root-product description of their recurrences, while a product of two finite Toeplitz sections differs from the finite section of the product by finite-rank corner corrections. We connect these structures by identifying multiplication-induced boundary defects with exterior-degree sectors of the determinant recurrence. To each polynomial core we associate recurrence polynomials for all exterior degrees. Multiplication of cores becomes a graded convolution via composed products, while complementary degrees are related by a scaled reciprocal duality. On the matrix side, one-sided factors produce a defect filtration whose successive pieces are exactly these sectors. For general two-sided factors, independently weighting the two corner corrections gives an interval filtration: increasing either defect order adds one adjacent sector, and the cumulative recurrences are generically minimal. The multiplication-induced same-corner minors used in these filtrations lie in the standard row-column state module, whereas cross-corner imbalance corresponds instead to Laurent recentering. Opposite Laurent recenterings realize the individual sectors directly, and for several factors the admissible recenterings form an explicit lattice polytope. The pentadiagonal case gives the basic $1+4+1$ decomposition. The resulting framework gives a factor-level description of how finite-section boundary effects generate the recurrence spectrum of structured banded Toeplitz products.

1 Introduction

The paper asks how multiplying Toeplitz symbols changes determinant recurrences when finite sections introduce corner defects. It couples those defects to exterior-degree recurrence sectors, yielding graded multiplication rules and defect filtrations.

1 Introduction

  • For two tridiagonal factors, the generic pentadiagonal recurrence has six modes, split as 1 + 4 + 1; the termwise product captures only the middle four.The two one-dimensional sectors supply the modes absent from the termwise product.Source: §1 · p. 2
  • Multiplication of polynomial cores gives a graded convolution of exterior-degree recurrence profiles, with pairwise mode products encoded by composed products.Complementary degrees are related by a scaled reciprocal duality.Source: §1 · p. 2

1.1 Relation to previous work and contributions

  • The contribution is an exact coupling of classical root-product recurrences with finite-section boundary defects, producing generically sharp defect filtrations and recentering descriptions of sectors.Root-product modes and finite-section product corrections are established structures; the paper identifies their relationship.Source: §1.1 · p. 3
  • The framework gives factor-by-factor, root-free formulas for assembling canonical determinant recurrences, but makes no numerical-stability or performance claims.For several factors, admissible cumulative defect indices form a bounded lattice polytope whose integer points index the sectors.Source: §1.1 · p. 3

2 Graded recurrence profiles

The paper assigns recurrence polynomials to each exterior degree of a polynomial core, relates complementary degrees by scaled reciprocity, and composes profiles across factorizations. The resulting sector polynomials describe how root-product modes combine.

2 Graded recurrence profiles

  • For each exterior degree, the recurrence polynomial has fixed-cardinality root products as its modes, and generically it is the minimal recurrence polynomial of the associated determinant sequence.The polynomial remains an annihilator under the stated nonzero endpoint-coefficient condition, although exceptional specializations may make it nonminimal.Source: §2 · p. 4

2.1 Complementary-sector duality

  • Complementary exterior degrees have scaled reciprocal canonical annihilators, an identity that also holds under nongeneric specializations with nonzero constant and leading coefficients.In the balanced self-dual case, the profile can be reconstructed from its components up to the middle degree.Source: §2.1 · pp. 5–6

2.2 Composed products

  • Composed products are associative and commutative at the level of root multisets and admit a resultant formula that avoids adjoining roots.A direct-sum companion-matrix decomposition also holds when the two core polynomials are coprime; coprimality is not needed for the root-multiset identity.Source: §2.2 · pp. 6–7
  • For a product of two cores, each sector chooses i roots from the first factor and k − i from the second, and the full degree-k recurrence polynomial is the product of those sector polynomials.Each sector polynomial is a composed product of the factor recurrence polynomials.Source: §2.2 · pp. 6–7

3 Products of finite Toeplitz sections

Finite Toeplitz sections multiply up to corrections at opposite corners. The paper uses these boundary defects to construct recurrence filtrations whose sectors match exterior-degree components, with generic minimality for the one-sided case.

3.1 Exact corner-defect formula

  • The finite-section product defect splits into corrections at opposite corners, supported in the first rows and columns on one side and the last rows and columns on the other.For sufficiently large sections, the supports are disjoint.Source: §3.1 · p. 8

3.2 Defect-order filtration and graded sectors

  • For one-sided factors, interpolating between the exact finite product and the clean Toeplitz section sorts determinants by the number of defect directions used.This matrix construction realizes graded sectors without changing the Laurent placement of the factors.Source: §3.2 · p. 9
  • Same-corner boundary minors belong to the standard row-column transfer state module because their row-column displacement balances separately at each end.This closure can fail for arbitrary bounded-boundary perturbations.Source: §3.2 · p. 9
  • For each one-sided defect order k, the cumulative recurrence is generically minimal, and its successive associated graded pieces are exactly the recurrence sectors.Each filtered sequence is a scalar matrix coefficient of the transfer restricted to sectors of degree at most k.Source: §3.2 · pp. 11–12
  • The jth associated graded piece can be viewed as the space of jth compound maps across the two one-sided factors.Defect order at most k retains compound orders 0 through k; the quotient isolates order k.Source: §3.2 · p. 14
  • For balanced factors with h = 2, the successive generic recurrence orders are 1, 5, and 6; for h = 3, they are 1, 10, 19, and 20.At the final defect order, the filtration reaches the full generic recurrence order.Source: §3.2 · p. 14

4 One-diagonal growth and the Pascal rule

One-diagonal growth adds a recurrence sector while introducing a rank-one finite-section defect; special boundary data can suppress the new sector in the distorted product sequence.

4 One-diagonal growth and the Pascal rule

  • Each finite-section defect from one-diagonal growth has rank at most one, and has rank one when its corresponding vector is nonzero.The upper and lower factors produce corrections at opposite boundaries.Source: §4 · p. 15
  • The canonical recurrence polynomials for the enlarged bands split by Pascal’s identities into inherited and adjacent-sector factors.The polynomial core changes by multiplication with a linear core, yielding two sectors.Source: §4 · pp. 15–16
  • With unit-diagonal normalization, the distorted enlarged-band determinant sequence equals the old sequence, suppressing the adjacent sector.The larger canonical annihilator is therefore nonminimal for this distorted sequence.Source: §4 · p. 16
  • For general one-column corrections, cofactor expansions introduce nonprincipal Toeplitz minors, so clean determinant shifts alone do not generally close the recurrence.The extreme case r = 1 closes immediately on clean Toeplitz determinants of different sizes.Source: §4 · p. 16
  • The Cauchy–Binet chain has an endpoint equal to the determinant sequence for the adjacent profile component.For upper growth, the last endpoint is the Toeplitz section of za.Source: §4 · p. 16
  • Filtering the inherited recurrence factor from the rank-one defect leaves an adjacent-sector sequence whose recurrence is generically minimal.The rank-one determinant identity gives the filtering relation.Source: §4 · p. 17
  • A zero-diagonal factor trades one lower diagonal for one upper diagonal, producing extreme recentering rather than genuine growth.In this case the polynomial core is only rescaled.Source: §4 · p. 18

5 Recentring and defect indices

Opposite Laurent shifts preserve the product symbol while changing the factors’ exterior degrees, realizing individual recurrence sectors through recentered finite-section products. With two-sided factors, left and right defect orders filter an interval of the same sector ladder, generically with minimal recurrences.

5 Recentring and defect indices

  • Opposite shifts of the factors preserve their product while changing which coefficients lie on the main diagonal.The shifted semibandwidths are (ra − δ, sa + δ) and (rb + δ, sb − δ), subject to both remaining finite Laurent bands.Source: §5 · p. 18
  • Increasing the recentering index by one transfers one unit of defect rank from the left corner to the right while keeping total defect rank fixed.This transfer defines the defect index.Source: §5 · p. 18
  • The defect-index factorization assigns a recurrence polynomial to each admissible recentered sequence, and the full determinant recurrence is assembled from these sector polynomials.Each sector polynomial is generically the minimal recurrence polynomial of its recentered sequence.Source: §5 · p. 19
  • Each sector is realized by a different recentered finite-section product, rather than as a cofactor term of one fixed product.The recentered products share the same Toeplitz bulk but allocate the fixed total boundary-defect rank differently.Source: §5 · p. 19

5.1 Two-corner interval filtration

  • With two-sided factors, defect orders k and ℓ select an interval of sectors −ℓ through k, and the corresponding recurrence is generically minimal.The bivariate filtration extends the existing recentering ladder from opposite ends rather than creating a mixed sector.Source: §5.1 · pp. 19, 21, 23, 24
  • The full interval recovers the canonical recurrence polynomial, while a zero corner rank reduces the construction to the one-sided filtration.The two boundary directions recover the same one-dimensional defect-index ladder as Laurent recentering.Source: §5.1 · p. 24

6 The quadratic-core pentadiagonal prototype

The pentadiagonal prototype shows how two quadratic determinant recurrences and boundary sectors combine to produce the generic degree-six recurrence.

6 The quadratic-core pentadiagonal prototype

  • The prototype’s purpose is to exhibit the canonical sector splitting of six generic modes, not to rederive existing five-diagonal determinant theory.The cited prior approaches include explicit formulas, generating functions, recurrence methods, and tridiagonal factorizations.Source: §6 · p. 24
  • For diagonal-corner corrections, selecting neither, either, or both corrections leaves clean determinant shifts Dn, Dn−1, or Dn−2.This closure on principal sections is special to diagonal-corner geometry.Source: §6 · p. 25
  • The resulting recurrence polynomial is Sweet’s order-six recurrence; hence the pentadiagonal recurrence can be reconstructed from two quadratic recurrences, their composed product, and two one-dimensional boundary sectors.Source: §6 · p. 26

7 Arbitrary products and the defect-index polytope

For several polynomial factors, admissible Laurent recenterings index determinant-recurrence sectors, and their cumulative shifts form a bounded defect-index polytope. Interface defect indices also encode the corner ranks at each factor cut.

7.1 Multi-index sectors

  • Each admissible multi-index recentering yields a determinant product annihilated by its sector recurrence polynomial, which is generically minimal.The sectors correspond to ways of distributing selected roots among the labelled factors.Source: §7.1 · p. 27
  • The multi-index sectors’ dimensions sum according to a multivariate Vandermonde identity.The number of sectors is separately obtained by coefficient extraction.Source: §7.1 · p. 28

7.2 Interface defect indices

  • An interface defect index is the net Laurent shift transported across the cut between two neighboring factors.The cumulative coordinates η_j are sums of the individual shifts δ_1 through δ_j.Source: §7.2 · p. 28
  • Admissible interface indices are integer points in a bounded polytope constrained by neighboring shifts and factor bandwidths.For more than two factors, neighboring defect indices are coupled, so their marginal intervals do not describe the full polytope.Source: §7.2 · pp. 28–29
  • The two corner defects at a factor cut have ranks determined by the recentering’s interface defect index.The cut calculation applies the corner-rank result to the shifted partial products on either side.Source: §7.2 · pp. 28–29
  • For three tridiagonal factors, the 20 generic modes split into eight central-sector modes and twelve modes across six nonzero-defect-index sectors.The seven sector dimensions sum to 20.Source: §7.2 · p. 29

8 A graded recurrence-profile algebra

The graded recurrence profiles of polynomial cores form an algebra under a multiplication induced by composed products. This structure supports recursive construction from a supplied factorization.

8 A graded recurrence-profile algebra

  • Multiplying polynomial cores corresponds to the graded convolution of their recurrence profiles, forming an associative, commutative multiplication with identity.The operation is induced by the classical composed product on root multisets.Source: §8 · pp. 29–30
  • Complete linear factorization resolves the sectors into individual S-subsets of roots, recovering the individual Widom modes.A single irreducible block instead treats the unperturbed determinant recurrence as one compound object.Source: §8 · p. 30
  • A supplied factorization allows the recurrence profile to be assembled recursively without extracting roots or constructing the full compound matrix.The appendix gives a soft-linear arithmetic bound in the degree of the expanded output recurrence.Source: §8 · p. 30

9 Concluding remarks

The paper connects finite-section boundary defects with graded determinant-recurrence sectors under structured Toeplitz multiplication. It gives factor-level descriptions of these recurrences, while leaving numerical stability and performance unclaimed.

9 Concluding remarks

  • Finite-section boundary defects under structured Toeplitz multiplication correspond to recurrence sectors, with one-sided filtrations and two-sided interval filtrations generically sharp.For several factors, cumulative defect indices form a lattice polytope whose integer points are the multi-index sectors.Source: §9 · p. 30
  • When a polynomial-core factorization is available, the canonical determinant recurrence can be assembled through factor-by-factor, root-free formulas.The paper presents these as potentially useful for symbolic and structured determinant computations, but makes no numerical-stability or performance claims.Source: §9 · p. 30
  • The matrix-side argument distinguishes same-corner minors in the standard row-column state module from cross-corner profile shifts and exceptional corner closures.Extensions to more general controlled finite-corner perturbations are identified as a direction for future work.Source: §9 · p. 30

Declarations

The authors disclose using generative AI during development and preparation, and state that they independently reviewed and verified the work.

Declarations

  • The authors used ChatGPT for exploratory mathematical reasoning, literature searches, organization, drafting, and editing, and independently reviewed and verified the final work.They state that they assume responsibility for all content.Source: Declarations · p. 31

Appendix A: Standard-state elimination of multiplication-induced corner defects

The appendix distinguishes balanced same-corner minors, which lie in the standard row-column state module, from cross-corner displacement, which changes the Laurent profile. It also describes generic elimination and an output-sensitive construction for already-factorized polynomial cores.

Appendix A: Standard-state elimination of multiplication-induced corner defects

  • Balanced same-corner multiplication minors belong to the standard row-column state module because their row and column deletions preserve boundary balance.Fixed-bandwidth elimination reduces them to normalized right-boundary states.Source: Appendix A · pp. 33–34
  • Cross-corner imbalance changes the Laurent profile rather than producing a state in the standard module for the original symbol.The endpoint Mn,n(a) = Dn(za) is identified as the same recentering mechanism in the one-diagonal calculation.Source: Appendix A · pp. 33–34
  • For generic coefficients, observability expresses a same-corner minor through consecutive principal determinants; at special coefficients, that observability step may fail.Standard-state membership still holds at specializations, but a short reduction then requires an additional identity.Source: Appendix A · pp. 34–35
  • Some boundary geometries admit shorter closures: terminal-chain cases reduce to principal sections, while the pentadiagonal identity selects Dn−1 or Dn−2.The deeper cofactors remain balanced same-corner minors, whereas the recentered endpoint does not.Source: Appendix A · pp. 35–36

Appendix B: Output-sensitive construction for factorized polynomial cores

  • The factorized-core construction assumes the factorization is supplied and bounded-degree, and its operation-count bound does not claim universal practical superiority.The cost of finding the factorization and bit complexity are outside the main bound.Source: Appendix B · p. 37

B.1 Pruning to the target exterior degree

The construction prunes each factor-tree node to degrees that can contribute to the target exterior degree, then bounds the retained work for balanced trees and treats edge degrees directly.

B.1 Pruning to the target exterior degree

  • At node v of degree dv, retain exactly degrees k with max(0, S − D + dv) ≤ k ≤ min(dv, S), since only these can extend to total degree S.The root retains only S, so its profile volume is N.Source: §B.1 · p. 37
  • The pruned factor-tree construction computes only retained child profiles, and the root polynomial is χQ,S.Complementary-sector duality can recover paired degrees without independently constructing each component.Source: §B.1 · pp. 37–38

B.2 Balanced trees

  • For θ-balanced factor trees under the theorem’s stated condition, total profile volume is linear in N.The complete dyadic-tree corollary gives an explicit bound for central target degrees.Source: §B.2 · pp. 39–40

B.3 The two edge degrees

  • For S = 1 and S = D − 1, the edge-degree polynomials have direct linear formulas and can both be constructed by coefficient rescaling in O(d) field operations.Source: §B.3 · pp. 40–41

B.4 Arithmetic complexity

For supplied bounded-degree factorizations, the appendix combines pruning and balanced factor trees to construct expanded recurrence annihilators without root extraction. Its profile-volume table illustrates intermediate-data savings, while the stated operation count excludes factorization and bit-complexity costs.

B.4 Arithmetic complexity

  • For a supplied factorization over a characteristic-zero field with bounded factor degrees, χQ,S can be computed in soft-linear field operations without root extraction or an N-dimensional auxiliary matrix.The edge degrees use direct formulas; complementary degrees can be recovered by duality.Source: §B.4 · pp. 41–42
  • The table compares internal profile volume with and without complementary-sector reconstruction; overall ratios approach 1 because both counts include the root output.The saving concerns non-root intermediate profile data.Source: §B.4 · p. 42, Table 1 · p. 42
  • The bound assumes a useful factorization is already available and does not control bit complexity or coefficient-height growth.The cost of finding the factorization may dominate, and output-height bounds are outside the paper’s scope.Source: §B.4 · p. 42
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