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for eta > 0 the images lie in I = [(-a - 2 beta - 2 sqrt(K))/d, (-a - 2 beta + 2 sqrt(K))/d] if and only if eta >= 1/2 and (s - (2 eta - 1) beta)^2 <= (2 eta - 1)^2 K, s = l - eta a (Theorem 2); with a >= 0, K >= beta^2, a + 2 beta >= 0, a^2 + 4c >= 0, l >= 0 and the scalar criterion, PF inputs with roots in [-R, 0], R = beta + sqrt(K), have PF images, sharply (the common-interlacer proof; the Chebyshev case is Iserles and Saff, Math. Proc. Cambridge Philos. Soc. 105 (1989), Theorem 1; the scalar criterion is RealRooted orderTwoConstants_nonneg_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"9453defebbe177b08bd7fa4ba9956ed37ac1393e3453938a3ba86b38514eb745","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-4"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-1bf99ad40f056d33","role":"sequence","name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"0"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]"}]},"repeat":true,"premises":[]},{"edge":{"id":"e-0178a3d1cd265fb6","operation":{"operation":"theorem_check","theorem":"ma_wang_interval"},"operation_detail":"ma_wang_interval","conclusion_id":"n-392f49b496f79fbf","premise_count":0,"theorem":"S.-M. Ma and Y. Wang, q-Eulerian polynomials and polynomials with only real zeros, Electron. J. Combin. 15 (2008) R17, Theorem 2 and Corollary 3 (arXiv:math/0611826), in the closed two-endpoint form of the RealRooted library (https://github.com/PerAlexandersson/RealRooted, RealRooted.affineEndpointStep_closed), after the change of variable x = U + (U - L) y, with the endpoint multiplicities tracked exactly through the induction","lean_route":null,"lean_declarations":["RealRooted.affineEndpointStep_closed","RealRooted.affineEndpointStep_roots_mem_Icc_iff","RealRooted.affineEndpointStep_rootMultiplicity_zero","RealRooted.affineEndpointStep_rootMultiplicity_neg_one","RealRooted.MaWang.endpointStep","RealRooted.MaWang.affineEndpointSequence","RealRooted.normalizedAffineEndpointSequence_roots_mem_Icc_iff"],"certificate_hash":"3693a8529cdd4207c93b80e3d7c9b1b6eca3d3cd48e6a136f374bba18e7adce0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-4"},"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]},"repeat":true,"premises":[]}]}]}]}]}],"truncated":false,"expanded":40}