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2017-04-07, 00:50   #2
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

26·131 Posts

Quote:
 Originally Posted by carpetpool Let n be an integer defining the cyclotomic properties of K (meaning that n is a factor of the cyclotomic polynomial C_K(x) evaluated at some x value). How many polynomials P(x), the same degree as C_K(x) their coefficients add up to n? For instance, choosing the cyclotomic field 3, C_3(x) = x^2+x+1, and x = 7, 19 is a factor of 7^2+7+1. How many polynomials P(x) of the form ax^2+bx+c defining the same field as x^2+x+1 is it the case that a+b+c = 19 where a, b, c are integers -19 <= (a, b, c) <= 19? Thanks for help, comments, and clarification.
well the number of polynomials total before the field consideration is 6*partitions(19,,[3,3]) ( as there are 6 orders possible for {a,b,c} ) or 180 ( okay I see now, you include negatives which throw the numbers off a bit but I was only trying to give a maximum).

-d,d,19 20*6 polynomials with ordering changes like this ( edited to include +0 and -0)
d,-(d-1),18 where d is positive, ....

etc.

edit2: turns out there are 400 possibilities to look through ( as some have only one order that is unique better than searching all 59319 {a,b,c} in that range by hand though.

edit 2 + :
Code:
my(a=[-19..19]);b=setbinop((x,y)->concat(x,y),a);b=setbinop((x,y)->concat(x,y),b,a);b=select(r->vecsum(r)==19,b)

Last fiddled with by science_man_88 on 2017-04-07 at 01:22