# Finding constraints on polynomials from their plot

Suppose I have the following list of rational functions (fraction of polynomials):

l={(353.073 x + 1142. x^2 + 2310.76 x^3 + 3770.26 x^4 + 5368.18 x^5 +
6304.07 x^6 + 6704.84 x^7 + 6513.96 x^8 + 5442.09 x^9 +
4620.29 x^10 + 3460.76 x^11 + 2654.19 x^12 + 2097.3 x^13 +
1517.29 x^14 + 1074.89 x^15 + 758.512 x^16 + 476.296 x^17 +
263.28 x^18 + 204.555 x^19 + 110.075 x^20 + 56.7865 x^21 +
27.375 x^22 + 28.2448 x^23 + 19.7196 x^24 + 5.61942 x^25 +
3.73647 x^26 + 0.931673 x^27 + 1.85847 x^28 + 2.7804 x^29 +
0.924375 x^30 + 1.84391 x^31)/(354 x + 1143 x^2 + 2320 x^3 +
3811 x^4 + 5441 x^5 + 6403 x^6 + 6829 x^7 + 6658 x^8 + 5571 x^9 +
4737 x^10 + 3560 x^11 + 2741 x^12 + 2174 x^13 + 1579 x^14 +
1120 x^15 + 789 x^16 + 502 x^17 + 275 x^18 + 215 x^19 + 117 x^20 +
59 x^21 + 30 x^22 + 30 x^23 + 21 x^24 + 6 x^25 + 4 x^26 + x^27 +
2 x^28 + 3 x^29 + x^30 + 2 x^31),(354.073 x + 1142. x^2 + 2310.76 x^3 + 3770.26 x^4 + 5368.18 x^5 +
6304.07 x^6 + 6704.84 x^7 + 6513.96 x^8 + 5442.09 x^9 +
4620.29 x^10 + 3460.76 x^11 + 2654.19 x^12 + 2097.3 x^13 +
1517.29 x^14 + 1074.89 x^15 + 758.512 x^16 + 476.296 x^17 +
263.28 x^18 + 204.555 x^19 + 110.075 x^20 + 56.7865 x^21 +
27.375 x^22 + 28.2448 x^23 + 19.7196 x^24 + 5.61942 x^25 +
3.73647 x^26 + 0.931673 x^27 + 1.85847 x^28 + 2.7804 x^29 +
0.924375 x^30 + 1.84391 x^31)/(354 x + 1143 x^2 + 2320 x^3 +
3811 x^4 + 5441 x^5 + 6403 x^6 + 6829 x^7 + 6658 x^8 + 5571 x^9 +
4737 x^10 + 3560 x^11 + 2741 x^12 + 2174 x^13 + 1579 x^14 +
1120 x^15 + 789 x^16 + 502 x^17 + 275 x^18 + 215 x^19 + 117 x^20 +
59 x^21 + 30 x^22 + 30 x^23 + 21 x^24 + 6 x^25 + 4 x^26 + x^27 +
2 x^28 + 3 x^29 + x^30 + 2 x^31),(354.071 x + 1146.97 x^2 + 2305.8 x^3 + 3799.95 x^4 + 5446.15 x^5 +
6375.93 x^6 + 6808.91 x^7 + 6485.56 x^8 + 5362. x^9 +
4555.02 x^10 + 3408.3 x^11 + 2624.15 x^12 + 2092.47 x^13 +
1510.54 x^14 + 1065.28 x^15 + 757.553 x^16 + 477.252 x^17 +
261.372 x^18 + 202.652 x^19 + 110.075 x^20 + 55.8401 x^21 +
28.3189 x^22 + 28.2448 x^23 + 19.7196 x^24 + 5.61942 x^25 +
3.73647 x^26 + 0.931673 x^27 + 1.85847 x^28 + 2.7804 x^29 +
0.924375 x^30 + 1.84391 x^31)/(354 x + 1143 x^2 + 2320 x^3 +
3811 x^4 + 5441 x^5 + 6403 x^6 + 6829 x^7 + 6658 x^8 + 5571 x^9 +
4737 x^10 + 3560 x^11 + 2741 x^12 + 2174 x^13 + 1579 x^14 +
1120 x^15 + 789 x^16 + 502 x^17 + 275 x^18 + 215 x^19 + 117 x^20 +
59 x^21 + 30 x^22 + 30 x^23 + 21 x^24 + 6 x^25 + 4 x^26 + x^27 +
2 x^28 + 3 x^29 + x^30 + 2 x^31),(352.071 x + 1180.79 x^2 + 2595.51 x^3 + 4326.4 x^4 + 5922.86 x^5 +
6615.14 x^6 + 6797.13 x^7 + 6102.68 x^8 + 5069. x^9 +
4241.35 x^10 + 3240.21 x^11 + 2532.09 x^12 + 2019.01 x^13 +
1455.59 x^14 + 1009.52 x^15 + 734.539 x^16 + 446.647 x^17 +
249.925 x^18 + 185.526 x^19 + 101.535 x^20 + 54.8936 x^21 +
27.375 x^22 + 28.2448 x^23 + 18.7806 x^24 + 5.61942 x^25 +
1.86824 x^26 + 1.86335 x^27 + 2.7877 x^28 + 1.8536 x^29 +
0.924375 x^30 + 0.921956 x^31)/(354 x + 1143 x^2 + 2320 x^3 +
3811 x^4 + 5441 x^5 + 6403 x^6 + 6829 x^7 + 6658 x^8 + 5571 x^9 +
4737 x^10 + 3560 x^11 + 2741 x^12 + 2174 x^13 + 1579 x^14 +
1120 x^15 + 789 x^16 + 502 x^17 + 275 x^18 + 215 x^19 + 117 x^20 +
59 x^21 + 30 x^22 + 30 x^23 + 21 x^24 + 6 x^25 + 4 x^26 + x^27 +
2 x^28 + 3 x^29 + x^30 + 2 x^31)}


They can be plotted using Plot[{1, l}, {x, 0, 1}, PlotRange -> All] I want to pick/select those polynomials that have lines above 1 at any given x how can I do this without counting the lines by hand? in this case the only element which will be dropped is l[[1]] as it is all below 1. I want to pick those line that all/some of them are above the line 1.

• It is unclear whether they must be "above 1", i.e., > 1; or "(not) below 1", i.e., >= 1. Assuming the latter, polys = Select[l, MaxValue[{#, 0 <= x <= 1}, x] >= 1 &] // Quiet; Commented Nov 21, 2018 at 16:38
• @BobHanlon they should start above 1 or have bits above 1 or start below 1 and for later x be above 1. Commented Nov 21, 2018 at 16:55
• Since you say "above 1" then change the criteria in my comment to > rather than >= Commented Nov 21, 2018 at 17:19

    Map[FindMaximum[{#, 0 <= x <= 1}, {x, 1}][[1]] &, l]