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

• To: mathgroup at smc.vnet.net
• Subject: [mg93311] Re: [mg93280] Trinomial decics x^10+ax+b = 0; Help with Mathematica code
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Mon, 3 Nov 2008 05:28:49 -0500 (EST)
• References: <200811020657.BAA02645@smc.vnet.net>

```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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