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

• To: mathgroup at smc.vnet.net
• Subject: [mg93324] 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 06:15:43 -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> <EDB4570E-5100-49D6-8E8B-D8FC3CAD3C39@mimuw.edu.pl> <E0EDBBCB-D8DA-402F-BF94-30FA77960273@mimuw.edu.pl>

```And one more thing. Don't forget the definition of h, which I failed
to include below:

In[2]:= h = Resultant[f, g, n]
Out[2]= m^45 + 246*a*m^36 - 502*b*m^35 - 13606*a^2*m^27 +
51954*a*b*m^26 - 73749*b^2*m^25 - 245*a^3*m^18 +
135060*a^2*b*m^17 - 92850*a*b^2*m^16 +
383750*b^3*m^15 + 13605*a^4*m^9 + 27200*a^3*b*m^8 +
25125*a^2*b^2*m^7 + 12500*a*b^3*m^6 + 3125*b^4*m^5 -
a^5

Andrzej Kozlowski

On 3 Nov 2008, at 17:11, Andrzej Kozlowski wrote:

> One more mistake (spurious semicolon). The correct version 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
>
>
> On 3 Nov 2008, at 17:07, Andrzej Kozlowski wrote:
>
>> 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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