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Hermite–Poulain: X·ψ(D) + α·ψ'(D) preserves real-rootedness - #2545
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…rooted ψ, α ≥ 0 Proved through the univariate Weyl-symbol criterion: the signed symbol x ψ(-y) + α ψ'(-y) does not vanish on the product of upper half-planes. PF addendum for nonnegative ψ, and corollaries θ + α, X D² + β D, X D^m + α m D^(m-1) and X D² + (X + 2α) D + α. Closes #2544. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
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Closes #2544.
New module
RealRooted/Hermite/PoulainSymbolDerivative.lean(209 lines):symbolDerivativeOperator_eq_zero_or_splits: for nonzero real-rootedψandα ≥ 0,X ψ(D) p + α ψ'(D) pis zero or real-rooted wheneverpis. No hypothesis onψ(0). Proof: the existing Weyl-symbol criterion (sum_mul_iterate_derivative_eq_zero_or_splits); the signed symbolx ψ(−y) + α ψ'(−y)is nonvanishing on the upper half-planes because the logarithmic derivative of a real-rooted polynomial has nonpositive imaginary part there.symbolDerivativeOperator_isPFPolynomial: PF preservation forψwith nonnegative coefficients.eulerShift_eq_zero_or_splits(θ + α),X_mul_derivative_derivative_add_eq_zero_or_splits(X D² + β D),X_mul_iterate_derivative_add_eq_zero_or_splits(X D^m + α m D^(m−1)),X_mul_derivative_derivative_add_affine_eq_zero_or_splits(X D² + (X + 2α) D + α).Checks: guarded
lake-workspace build RealRooted.Hermite.PoulainSymbolDerivative RealRooted.Productionrc=0, no warnings in the new module; source guards pass.🤖 Generated with Claude Code