Problem of the Month (June 2018)

Consider polynomials of order n–1 with coefficients that are some permutation of 1, 2, 3, ... n. How many and what kind of roots can they have? In particular, what is the max/min number of integer/real/complex roots they can have, distinct/not? Which of these polynomials is straightest on the interval [–1,1] (measured by minimizing the maximum curvature)? Which of these polynomials that is non-negative on [–1,1] has smallest area on the interval [–1,1]? What other questions involving these polynomials are interesting?

ANSWERS

The best known solutions are shown below.

Maximum Integer Roots
n# RootsRootsCoefficientsSolutionsAuthor
10 11
21–21 21
32–1, –21 3 21
41–11 2 4 38
51–21 3 4 5 22
61–21 3 4 6 5 29
72–1, –11 2 6 7 3 5 426
83–1, –1, –23 7 2 4 8 1 5 61
92–2, –21 5 9 7 2 3 6 4 82
10010!Johan de Ruiter
112–1, –21 2 3 6 4 11 7 5 8 9 10786Johan de Ruiter
122–1, –21 2 3 7 6 11 9 10 8 5 12 43132Johan de Ruiter
13013!Johan de Ruiter
141–21 2 3 5 4 12 6 13 7 11 8 14 9 101947354Johan de Ruiter

Maximum Real Roots
n# RootsRootsCoefficientsSolutionsAuthor
10 11
21–21 22
32–1, –21 3 22
41–11 2 4 324
52–2.524, –1.1941 3 2 4 538
63–3.414, –0.682, –0.5851 4 3 5 6 214
74–3.310, –1, –1, –0.8651 5 6 2 4 7 34
83–1.514, –1, –11 2 3 7 6 4 8 52282
94–2.582, –1.526, –0.817, –0.2961 4 5 6 7 3 8 9 2448
105–3.317, –2.023, –1.262, –0.571, –0.4531 6 10 4 3 7 5 9 8 24
114–3.247, –1, –0.797, –0.2961 4 5 11 10 6 7 3 8 9 2?Maurizio Morandi
125–3.213, –2.412, –1.323, –0.859, –0.1861 6 10 5 8 9 4 11 3 7 12 2?Maurizio Morandi

Maximum Repeated Complex Roots
n# Repeated RootsRepeated RootsCoefficientsSolutions
10 11
20 1 22
30 1 2 36
40 1 2 3 4 24
50 1 2 3 4 5120
60 1 2 3 4 5 6720
71–11 2 6 7 3 5 426
81–11 2 3 7 6 4 8 5120
91–21 5 9 7 2 3 6 4 84
102(1+√3)/2, (1–√3)/21 6 2 9 7 10 8 4 5 36

Minimum Curvature on [–1,1]
nCurvatureCoefficientsSolutionsAuthor
1011
201 22
30.707+1 3 21
40.041+2 1 4 31
50.034+1 3 2 5 41
60.0248+3 2 4 1 6 51
70.0181+5 6 1 4 2 7 31
80.01310+2 6 5 1 4 3 8 7?
90.01017+4 8 6 2 1 5 3 9 7?Maurizio Morandi
100.007993+4 8 7 6 2 1 5 3 10 9?Maurizio Morandi
110.006582+5 9 8 7 4 2 1 6 3 11 10?Maurizio Morandi
120.005503+5 10 9 8 7 4 2 1 6 3 12 11?Maurizio Morandi

Minimum Area on [–1,1]
nAreaCoefficientsSolutionsAuthor
1211
241 21
343 2 11
414/3 = 4.666+2 4 3 12
528/5 = 5.64 5 3 2 11
6106/15 = 7.066+2 6 5 4 3 14
738/5 = 7.67 6 4 5 3 2 11
8304/35 = 8.685+4 8 7 6 5 3 2 14
9598/63 = 9.492+8 9 6 7 5 4 3 2 11Maurizio Morandi
10680/63 = 10.793+8 10 7 9 6 5 4 3 2 110Maurizio Morandi
11888/77 = 11.532+10 11 9 8 6 7 5 4 3 2 11Maurizio Morandi
1243354/3465 = 12.511+10 12 9 11 7 8 6 4 5 3 2 12Maurizio Morandi

Maurizio Morandi proved that the minimum area must be at least n.

Maurizio Morandi thought we should investigate the polynomials which minimize arclength as well.

Minimum Arclength on [–1,1]
nArclengthCoefficientsSolutionsAuthor
1211Maurizio Morandi
22.828+1 21Maurizio Morandi
34.646+1 2 31Maurizio Morandi
47.218+2 3 1 41Maurizio Morandi
511.169+2 4 3 1 51
616.509+1 3 5 4 2 61
722.486+1 3 5 6 4 2 71
829.128+2 6 7 5 4 3 1 81
937.130+2 6 8 7 5 4 3 1 91Maurizio Morandi
1046.482+1 3 5 8 9 7 6 4 2 101Maurizio Morandi
1156.495+1 3 5 8 10 9 7 6 4 2 111Maurizio Morandi
1267.136+2 6 9 11 10 8 7 5 4 3 1 121Maurizio Morandi

Maurizio Morandi proved that the minimum arclength must be at least (n2–n+2)/2.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 6/1/18.