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Veronese sections preserve constant-coefficient linear recurrences (#2533 part 2) - #2546
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Over any commutative ring, every r-section of a solution of an order-d LinearRecurrence over A[X] satisfies one recurrence of order r*d whose characteristic polynomial is the explicit section-matrix determinant veroneseNorm of the original one (adjugate factorization; no roots of unity). Eventual-start version, real bridge to veroneseSectionPolynomial, a real-rootedness corollary, three breadth examples and catalog entries. Part 2 of #2533. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
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Part 2 of #2533, in the agreed form (Mathlib
LinearRecurrence; explicit determinant norm).Generic (Mathlib-only, upstreamable):
RealRooted/Mathlib/Algebra/Polynomial/Veronese.lean(464 lines):Polynomial.veroneseSection,veroneseSection_expand_mul, the section multiplication matrix,veroneseNorm r p = det,natDegree_veroneseNorm_le(≤ r·deg),coeff_veroneseNorm_zero,Monic.veroneseNorm,mul_veroneseAdjugateCoeffs(p · Q' = expand r (veroneseNorm r p)).RealRooted/Mathlib/Algebra/LinearRecurrence/Veronese.lean(130 lines):LinearRecurrence.veronese,veronese_order,charPoly_veronese,IsSolution.veronese,exists_veroneseSection,veronese_recurrence_of_eventually(start N + (r−1)d).Real API:
veroneseSectionPolynomial_eq_veroneseSection(bridge; existing signatures unchanged) andisSolution_and_eq_zero_or_splits_veroneseSectionPolynomial.Breadth (
Tactic/Examples/VeroneseRecurrence.lean): binomial rows r = 2 (order 2), Fibonacci tiling rows r = 3 (order 6), Delannoy rows r = 2 (order 4), each by applying the general theorem. The worked examples were checked by exact computation to n = 30 (issue comment).Catalog: Veronese sections page gains the definitions and theorems. Import budgets: Hadamard closure 354 → 358, broad umbrella +1 module each for the new files.
Checks: guarded
lake-workspace build RealRooted RealRooted.Production RealRooted.Tactic.Examples RealRooted.Challenges.PolyaFrequency.VeroneseSectionsrc=0, no warnings; all source guards and catalog validation pass.🤖 Generated with Claude Code