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. >>>> >>>> But my skills with Mathematica coding is very limited. Please help. >>>> >>>> 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 >>>> >>> >> >

**References**:**Trinomial decics x^10+ax+b = 0; Help with Mathematica code***From:*tpiezas@gmail.com