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In a recent post to which I had given a solution, I had not been able to prove in a rigorous way (i.e., without any recourse to any approximation) that polynomial:

$$f(x) = 4 - 8\,x + 4\,x^2 + 4\,x^4 - 6\,x^5 - 4\,x^6 + 6\,x^7 + 4\,x^8 - x^{10} - 4\,x^{11} - x^{12} + 2\,x^{13} + x^{14}\\ $$

has no real roots, which amounts to say that $\forall x, \ f(x)>0$.

(the curve of $f$ is visibly above $x$ axis for all $x$, as can be seen on the figure, but I am looking for "clean" arguments...).

I have made several attempts, especially by trying to find simpler polynomial expressions $g(x)$ such that $g(x) \leq f(x)$, to use complex variable techniques (Rouché's theorem...), etc. unsuccessfully.

Has somebody a solution ? enter image description here

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Jean Marie
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  • The minimum value of this very high degree polynomial is $\leq \frac{1}{2}$, so one doesn't have much room at all here. – Travis Willse May 13 '16 at 12:42
  • @Travis Yes, and the derivative is not factorizable... but meanwhile loup blanc has given a solution using a Sturm sequence. – Jean Marie May 13 '16 at 13:35
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    It is sad that on Math StackExchange, some people (may be always the same) downvote questions (and even propose to close them) in an anonymous manner. Maybe, in the present case, he/she/they has/have not understood the issue; then, why not ask precisions in a preliminary step... – Jean Marie May 13 '16 at 13:42
  • Yes, I suspect that something like loup blanc's solution or something similarly computationally intensive might be necessary. Notice that it follows from making suitable linear substitutions and observing the simple sign changes of the terms (or rather, the lack thereof), we can see that any real solution must be in $[-1, 1]$, and perhaps this might lead to something doable by hand. – Travis Willse May 13 '16 at 14:03

1 Answers1

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It suffices to write the Sturm's sequence, cf. "Sturm's theorem" in

https://en.wikipedia.org/wiki/Sturm's_theorem

or rather, to ask Maple or Mathematica to write it. It is a formal method (as a gcd calculation); therefore the calculations can be huge.

Here Maple gives

$\left[{x}^{14}+2\,{x}^{13}-{x}^{12}-4\,{x}^{11}-{x}^{10}+4\,{x}^{8}+6\,{x}^ {7}-4\,{x}^{6}-6\,{x}^{5}+4\,{x}^{4}+4\,{x}^{2}-8\,x+4,{x}^{13}+{ \frac {13}{7}}\,{x}^{12}-6/7\,{x}^{11}-{\frac {22}{7}}\,{x}^{10}-5/7\, {x}^{9}+{\frac {16}{7}}\,{x}^{7}+3\,{x}^{6}-{\frac {12}{7}}\,{x}^{5}-{ \frac {15}{7}}\,{x}^{4}+{\frac {8}{7}}\,{x}^{3}+4/7\,x-4/7,-10+{x}^{12 }+9/5\,{x}^{11}-2/5\,{x}^{10}-{\frac {21}{5}}\,{x}^{8}-{\frac {131}{20 }}\,{x}^{7}+{\frac {133}{20}}\,{x}^{6}+{\frac {177}{20}}\,{x}^{5}-{ \frac {31}{4}}\,{x}^{4}-{\frac {42}{5}}\,{x}^{2}+{\frac {92}{5}}\,x-1/ 4\,{x}^{9}+2/5\,{x}^{3},-{\frac {25}{4}}\,{x}^{9}-17\,{x}^{3}+{x}^{11} +{\frac {41}{8}}\,{x}^{10}-{\frac {97}{8}}\,{x}^{8}+{\frac {57}{8}}\,{ x}^{7}+{\frac {89}{8}}\,{x}^{6}-{\frac {79}{8}}\,{x}^{5}+{\frac {15}{4 }}\,{x}^{4}+32\,{x}^{2}-17\,x,{\frac {128}{293}}-{x}^{10}+{\frac {661} {293}}\,{x}^{8}-{\frac {77}{293}}\,{x}^{7}-{\frac {685}{293}}\,{x}^{6} +{\frac {355}{293}}\,{x}^{5}-{\frac {278}{293}}\,{x}^{4}-{\frac {1472} {293}}\,{x}^{2}+{\frac {488}{293}}\,x+{\frac {114}{293}}\,{x}^{9}+{ \frac {1128}{293}}\,{x}^{3},{x}^{9}+{\frac {34152}{79351}}\,{x}^{3}-{ \frac {2224}{79351}}\,{x}^{8}-{\frac {143283}{79351}}\,{x}^{7}+{\frac {23809}{79351}}\,{x}^{6}+{\frac {177833}{79351}}\,{x}^{5}-{\frac { 101647}{79351}}\,{x}^{4}-{\frac {255976}{79351}}\,{x}^{2}+{\frac { 316752}{79351}}\,x-{\frac {103400}{79351}},-{\frac {4874668}{2473535}} -{x}^{8}-{\frac {3702696}{2473535}}\,{x}^{7}+{\frac {1102024}{2473535} }\,{x}^{6}+{\frac {4719882}{2473535}}\,{x}^{5}+{\frac {13061}{107545}} \,{x}^{4}-{\frac {712376}{2473535}}\,{x}^{2}+{\frac {5795756}{2473535} }\,x-{\frac {2517376}{2473535}}\,{x}^{3},-{\frac {130644976}{71135621} }\,{x}^{3}-{x}^{7}-{\frac {117877931}{71135621}}\,{x}^{6}+{\frac { 42199772}{71135621}}\,{x}^{5}+{\frac {191521943}{71135621}}\,{x}^{4}+{ \frac {34202204}{71135621}}\,{x}^{2}+{\frac {119672796}{71135621}}\,x- {\frac {131248248}{71135621}},{\frac {4636190692}{845120667}}+{x}^{6}+ {\frac {469969805}{281706889}}\,{x}^{5}-{\frac {4887791734}{845120667} }\,{x}^{4}+{\frac {1291091456}{281706889}}\,{x}^{2}-{\frac {9119197856 }{845120667}}\,x+{\frac {3667395556}{845120667}}\,{x}^{3},-{\frac { 118807258}{229387925}}\,{x}^{3}+{x}^{5}-{\frac {23751047557}{ 17433482300}}\,{x}^{4}+{\frac {1733801327}{871674115}}\,{x}^{2}-{ \frac {6098757498}{4358370575}}\,x+{\frac {1595181704}{4358370575}},-{ \frac {14165381246176}{3678995429977}}+{x}^{4}+{\frac {148022761860}{ 3678995429977}}\,{x}^{2}+{\frac {77449670408}{12730087993}}\,x-{\frac {12687975462712}{3678995429977}}\,{x}^{3},-{x}^{3}+{\frac { 67591683841151}{107354894758164}}\,{x}^{2}+{\frac {82833966032893}{ 53677447379082}}\,x-{\frac {543419063876153}{429419579032656}},{\frac {3543406427639359}{2400254723765948}}+{x}^{2}-{\frac {1466258753587345 }{600063680941487}}\,x,x-{\frac {75792061996548545}{75819308999515908} },-1\right]$

Travis Willse
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  • Thanks very much, I had heard about that method a long time ago but never used it (thus hadn't the idea to use it), and didn't know that computer algebra systems had this method. – Jean Marie May 13 '16 at 13:44
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    I've taken the liberty of converting the Maple output to $\LaTeX$ using the Maple command $\texttt{latex()}$ (as you can see, it seems to treat fractions with single-digit entries differently). – Travis Willse May 13 '16 at 13:58
  • I thought about a method but it had a flaw in it because I assumed implicitly that the function is convex. – Jean Marie May 13 '16 at 21:13