Re: picking coefficients
- To: mathgroup at smc.vnet.net
- Subject: [mg118169] Re: picking coefficients
- From: Peter Pein <petsie at dordos.net>
- Date: Fri, 15 Apr 2011 03:55:59 -0400 (EDT)
- References: <io6d3s$drp$1@smc.vnet.net>
Am 14.04.2011 11:00, schrieb Kent Holing:
> I have a general polynomial in 4 variables a,b,c,d.
>
>> From all terms N a^i b^j c^k d^l for i,j,k,l positive integers (all or some of i,j,k,l may be 0) with N a numerical (integer) coefficient ( N/= 0, N may be negative or positive) of the polynomial, I want to pick among these terms only those where N /== 0 mod 8.
>
> Is it an easy way to achieve this, using Mathematica?
>
> Example/testcase:
> For Q(x) = x^4 + (2a+1) x^3 + 2b x^2 + 2c x + 2d + 1 = 0
> where a, b, c and d are integers, let the polynomial
> be the discriminant of the quartic Q(x) = 0.
>
> The result of the above request should then be 5 + 4a(a+1) +4b(b+1) + 4c(c+1) + 4d(d+1), showing that the discriminant == 5 mod 8 and therefore not a square.
>
> In fact, this shows that the Galois group of the quartic must be either Z4, D4 or S4.
>
> Kent Holing
>
Hi Kent,
I don't know how to get the wanted result (experimenting gave just 5 + 4
d (d+1) ), but the final result can be obtained by typing simply:
Simplify[
Mod[Discriminant[x^4 + (2a+1) x^3 + 2b x^2 + 2c x + 2d + 1, x], 8],
Element[{a, b, c, d}, Integers]]
Peter