{"chains":[{"edge":{"id":"e-3f86cc05e6d83baf","operation":{"index_shift":2,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"-2\",\"intercept\":\"0\"}"},"operation_detail":"affine, shift 2","conclusion_id":"n-aceeb2572b6504c7","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"b21285a6fa36182c5459d95869afd26c8c6537222feef368d0558e0878aea40f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-be41df8499a2681b","role":"sequence","name":"Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].","uses":{"claim":"real_rooted","from_index":3,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 3, is zero or real-rooted with all roots in (-inf, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-be41df8499a2681b","role":"sequence","name":"Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].","uses":{"claim":"real_rooted","from_index":3,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 3, is zero or real-rooted with all roots in (-inf, 0]"},"chains":[{"edge":{"id":"e-a903e57fab33046a","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-be41df8499a2681b","premise_count":0,"theorem":"Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; S_n = A + B D + C D^2 with A = 2 + x, B = 8 + 8 x, C = 16 x: Branden's Weyl step: T = A + B D + C D^2 maps every real-rooted polynomial to a real-rooted polynomial when (i) B^2 - 4 A C >= 0 on the real line, (ii) A strictly interlaces B (deg B = deg A + 1) or strictly alternates left of B (deg B = deg A), with leading coefficients of the same sign when deg A >= 1, and (iii) T(x^k) has degree deg A + k with leading coefficients of one sign for every k >= 0 (P. Branden, Linear transformations preserving the Polya frequency property, Trans. Amer. Math. Soc. 358 (2006), Corollary 3.5); (i) is necessary for a real-rootedness preserver (J. Borcea and P. Branden, Proc. London Math. Soc. 101 (2010), Theorem 1.3); with coefficients of one sign T also keeps every root in (-inf, 0]; a linear operator preserving real-rootedness preserves the alternation of two polynomials (P. Branden, Trans. Amer. Math. Soc. 358 (2006), Corollary 2.5)","lean_route":null,"lean_declarations":[],"certificate_hash":"78c12b479cd2fd9904f099e93f133b6007c9ee1dc784db60d5db9e6895dd1ac1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]},{"edge":{"id":"e-f7a0355494cd983e","operation":{"operation":"theorem_check","theorem":"hypergeometric_row"},"operation_detail":"hypergeometric_row","conclusion_id":"n-be41df8499a2681b","premise_count":0,"theorem":"Row-hypergeometric rows: an exact row relation p(n, k) [x^k] P_n + q(n, k) [x^(k-1)] P_n = 0, proved for every n by an invariant row-annihilator system of the rule, writes every row as U_0(n) x^v (or its reversal) times the image of (1 + x)^M(n) under a product of nonnegative multiplier sequences: C^k (C > 0); k -> M'!/(M' - k)!, whose exponential generating function is (1 + x)^M' (Polya--Schur, Theorem 7.1 (3) => (1) of P. Branden, Unimodality, log-concavity, real-rootedness and beyond, arXiv:1410.6601: a sequence is a multiplier sequence when G(x) = sum_k lambda_k x^k / k! or G(-x) can be written C x^m e^(a x) prod_k (1 + alpha_k x) with a >= 0, alpha_k >= 0); k -> phi(k) for a polynomial phi with only real roots, all <= 0 (Laguerre's theorem, RealRooted isPFMultiplierSequence_eval_of_roots_nonpos); k -> 1/(alpha)_k for alpha > 0 (RealRooted isPFMultiplierSequence_inv_ascPochhammer, via the generalized Laguerre polynomials); products of PF multiplier sequences are PF multiplier sequences (RealRooted IsPFMultiplierSequence.mul) and map real-rooted polynomials with nonnegative coefficients to such polynomials (IsPFMultiplierSequence.diagonalOperator_isPF)","lean_route":null,"lean_declarations":["RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_inv_ascPochhammer","RealRooted.IsPFMultiplierSequence.mul","RealRooted.IsPFMultiplierSequence.diagonalOperator_isPF"],"certificate_hash":"553eedb638a72ba064b7f38a34c70210c1356072e15f24d0591ebb4a3e1e82a5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}}],"premises":[]},"premises":[]}]}]}],"truncated":false,"expanded":3}