       Re: Trinomial decics x^10+ax+b = 0; Help with Mathematica code

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• Subject: [mg93322] Re: [mg93280] Trinomial decics x^10+ax+b = 0; Help with Mathematica code
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Tue, 4 Nov 2008 06:15:20 -0500 (EST)
• References: <200811020657.BAA02645@smc.vnet.net> <BB6FC818-9D69-4DA2-8F6A-0A5B680ECD2C@mimuw.edu.pl> <64fc87850811020944p49edb6a2u38aa53a29b539784@mail.gmail.com> <F85E373D-0B9C-4090-BA7B-AE95575625E1@mimuw.edu.pl>

```Sorry, I forgot that one has to Catch whatever might be Thrown ;-)

Unfortunately, its unlikely to make any difference :-(

Anyway, the correct code is:

Catch[NestWhile[({a, b} = RandomInteger[{-10^6, 10^6}, {2}];
If[ck[a] == ck[b] == 0 && (p[a, b] || p[b, a]), Throw[{a, b}],
ck[a] = ck[b] = 1]) &, 1, True &, 1, 10^6]];//Timing

Andrzej Kozlowski

3 Nov 2008, at 16:08, Andrzej Kozlowski wrote:

> I doubt that you will have much luck with this approach. I did:
>
> f = -a + m^9 - 8 m^7 n + 21 m^5 n^2 - 20 m^3 n^3 + 5 m n^4 ; g = -b
> + m^8 n -
>  7 m^6 n^2 + 15 m^4 n^3 - 10 m^2 n^4 + n^5;
>
> p[x_, y_] :=
> MemberQ[Exponent[FactorList[h /. {a :> x, b :> y}][[All, 1]], m], 5
> | 10]
>
> (mm = Outer[List, Range[10^3], Range[10^3]]); // Timing
> {0.012557, Null}
>
> (vv = Apply[p, mm, {2}];) // Timing
> {7077.94, Null}
>
> Position[vv, True]
> {}
>
> There is not a single example among the first 10^6. You can try this
> approach on larger numbers, though eventually you will find yourself
> short of memory to story such large arrays. Or you could try random
> searches, with very much larger numbers, e.g.
>
> ck[_] = 0;
>
> NestWhile[({a, b} = RandomInteger[{-10^6, 10^6}, {2}];
>     If[ck[a] == ck[b] == 0 && (p[a, b] || p[b, a]), Throw[{a, b}],
>      ck[a] = ck[b] = 1]) &, 1, True &, 1, 10^4]; // Timing
>
> {0.481902, Null}
>
> As you see, 10^4 random pairs between -10^6 and 10^6, and no result.
> The "flag" ck, by the way, is used to mark the numbers we have
> already tried, so when we run this again we do not apply p to them
> again.
>
> Anyway, it looks to me much harder than the proverbial looking for a
> needle in a haystack.
>
> Andrzej Kozlowski
>
>
>
> On 3 Nov 2008, at 02:44, Tito Piezas wrote:
>
>> Hello Andrzej,
>>
>> The simple trinomials x^6+3x+3 = 0 and x^8+9x+9 = 0 are solvable,
>> both factoring over Sqrt[-3].
>>
>> However, there seem to be no known decic trinomials,
>>
>> x^10+ax+b = 0
>>
>> such that it factors over a square root (or quintic) extension.
>>
>> The 45-deg resultant, for some {a,b}, IF it has an irreducible 5th
>> or 10th degree factor, will give the {a,b} of such a decic.
>>
>> To find one, what I did was to manually substitute one variable,
>> like "a" (yes, poor me), use the Table[] function for "b" (as well
>> as the Factor[] function), and inspect the 45-deg to see if it has
>> the necessary 5th or 10th deg factors. Unfortunately, there are
>> none for {a,|b|} < 30.
>>
>> So what I was thinking is to extend the range to {a,|b|} < 1000.
>> But that's about 10^6 cases, far more than I could do by hand.
>>
>> I'm sure there is a code such that we only have to give the upper
>> bound for {a,b}, let Mathematica run for a while, and it prints out
>> ONLY the {a,b} such that the 45-deg has an irreducible 5th (or
>> 10th) factor. I believe there might be a few {a,b} -- but doing it
>> by hand is like looking for a needle in a haystack.
>>
>>
>> P.S. The 45-deg resultant should be in the variable "m", so pls
>> eliminate the variable "n" (not m) between the two original
>> equations.
>>
>> Sincerely,
>>
>> Tito
>>
>>
>> On Sun, Nov 2, 2008 at 6:14 AM, Andrzej Kozlowski
>> <akoz at mimuw.edu.pl> wrote:
>>
>> On 2 Nov 2008, at 15:57, tpiezas at gmail.com wrote:
>>
>> Hello guys,
>>
>> I need some help with Mathematica code.
>>
>> It is easy to eliminate "n" between the two eqn:
>>
>> -a + m^9 - 8m^7n + 21m^5n^2 - 20m^3n^3 + 5mn^4 = 0
>> -b + m^8n - 7m^6n^2 + 15m^4n^3 - 10m^2n^4 + n^5 = 0
>>
>> using the Resultant[] command to find the rather simple 45-deg
>> polynomial in "m", call it R(m).
>>
>> As Mathematica runs through integral values of {a,b}, if for some
>> {a,b} the poly R(m) factors, we are interested in two cases:
>>
>> Case1: an irreducible decic factor
>> Case2: an irreducible quintic factor
>>
>> What is the Mathematica code that tells us what {a,b} gives Case 1 or
>> Case 2?
>>
>>
>> Thanks.  :-)
>>
>>
>> Tito
>>
>>
>>
>>
>> Let f = -a + m^9 - 8 m^7 n + 21 m^5 n^2 - 20 m^3 n^3 + 5 m n^4 ; g
>> = -b + m^8 n -
>> 7 m^6 n^2 + 15 m^4 n^3 - 10 m^2 n^4 + n^5;
>>
>> and
>>
>> h = Resultant[f, g, m];
>> Exponent[h, n]
>> 45
>>
>> so h is a polynomial of degree 45. Now, let
>>
>> p[x_, y_] := Exponent[FactorList[h /. {a :> x, b :> y}][[All, 1]], n]
>>
>> computing p[a,b]  gives you the exponents of the irreducible factor
>> for the given values of a and b. In most cases you get {0,45} - the
>> irreducible case. But, for example,
>> p[a, 0]
>> {0, 1, 9}
>> and
>> p[2, 1]
>> {0, 9, 36}
>>
>> So now you can search for the cases you wanted.  You did not
>> seriously expect Mathematica would do it by itself, I hope?
>>
>> Andrzej Kozlowski
>>
>

```

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