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c > 0 and eta = m/d >= 0, every real polynomial with all roots in E = [-beta - sqrt(K), -beta + sqrt(K)] has a real-rooted image, and no interval of positive length outside E works (Theorem 1); 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. 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Soc. 105 (1989), Theorem 1; the scalar criterion is RealRooted orderTwoConstants_nonneg_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"475022965e77f29ce7d319a8938390b37d95f933ae3043d0d6cdf4fe637c8651","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"-2"},"value":"-1564556493/268435456"},"upper":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"2"},"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-07af2e6e3c9f0c77","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-2"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-2, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:        1;\n  1:        5,         2;\n  2:       31,        30,         5;\n  3:      215,       352,       133,        14;\n  4:     1597,      3780,      2366,       546,        42;\n  5:    12425,     38918,     35190,     13170,      2178,       132;\n  6:    99955,    391674,    472719,    252252,     66693,      8580,      429;\n  7:   824675,   3890108,   5948855,   4205410,   1568853,    319462,    33605,    1430;\n  8:  6939769,  38328840,  71566396,  63866660,  31327296,   8930350,  1476046,  131274,   4862;\n  9: 59334605, 375784138, 833321164, 907052692, 558693408, 208118586, 47866390, 6651658, 512278, 16796;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+3*n^2-5*n-6)*P(n,x) = -(2*n^3-9*n^2+7*n+6)*P(n-4,x) + (32*n^3-96*n^2+40*n+48+(8*n^3-24*n^2+10*n+12)*x)*P(n-3,x) - (124*n^3-186*n^2-34*n+48+(16*n^3-24*n^2-4*n+6)*x)*P(n-2,x) + (32*n^3-56*n-24+(8*n^3-14*n-6)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 5 + 2*x, P(2,x) = 31 + 30*x + 5*x^2, P(3,x) = 215 + 352*x + 133*x^2 + 14*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 2*x;\nP[2, x_] := 31 + 30*x + 5*x^2;\nP[3, x_] := 215 + 352*x + 133*x^2 + 14*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 9*n^2 + 7*n + 6)*P[n - 4, x] + (32*n^3 - 96*n^2 + 40*n + 48 + (8*n^3 - 24*n^2 + 10*n + 12)*x)*P[n - 3, x] - (124*n^3 - 186*n^2 - 34*n + 48 + (16*n^3 - 24*n^2 - 4*n + 6)*x)*P[n - 2, x] + (32*n^3 - 56*n - 24 + (8*n^3 - 14*n - 6)*x)*P[n - 1, x])/(2*n^3 + 3*n^2 - 5*n - 6)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 2*x; v[3] = 31 + 30*x + 5*x^2; v[4] = 215 + 352*x + 133*x^2 + 14*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(2*n^3 - 9*n^2 + 7*n + 6)*v[i - 4] + (32*n^3 - 96*n^2 + 40*n + 48 + (8*n^3 - 24*n^2 + 10*n + 12)*x)*v[i - 3] - (124*n^3 - 186*n^2 - 34*n + 48 + (16*n^3 - 24*n^2 - 4*n + 6)*x)*v[i - 2] + (32*n^3 - 56*n - 24 + (8*n^3 - 14*n - 6)*x)*v[i - 1])/(2*n^3 + 3*n^2 - 5*n - 6)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-66e2c3448245b0b7","steps":["x \u21a6 x+1","basis A008288"],"edges":["e-190c283dd8b47413","e-7a79b89fdfd077a7"],"text":"A033282 \u2192 x \u21a6 x+1 \u2192 basis A008288"},"headline":"A033282 \u2192 x \u21a6 x+1 \u2192 basis A008288","chain":null}