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I want to find Resultant of trinomials of this form $x^i-x^j-1$. Is it possible to do symbolic calculation for this in sage ?

Consider the following code :-

p=10010113
F=GF(p)
R=PolynomialRing(F,'x')
f=x^5 + 3212480*x^4 + 5943978*x^3 + 1041193*x^2 + 3212605*x + 4505026
F1.<a>=F.extension(f)
R1=PolynomialRing(F1,'x')
f1=derivative(f(x),x)
b=R1(f1(a))
b=F1(b)

Now since b $in $ F1 . Therefore F1(b**p^5-1) should output 1, but I am getting this output

sage: F1(b**(((p^5)-1)))
/usr/lib/sagemath/local/bin/sage-ipython:1: RuntimeWarning: invalid value encountered in power
  #!/usr/bin/env python
9745575*a^4 + 8100949*a^3 + 6855548*a^2 + 351457*a + 548263

Which is absurd !

Are we replacing $y$ by $x^2+ax+b$ everywhere ?

I evaluated Divison polynomials using

R.<A,B>=PolynomialRing(ZZ)
E = EllipticCurve([A,B])
g = E.division_polynomial(k)

The results i noticed were just function of $x$ , In theory i saw that division polynomials for k even depends on y. In there something I am unable to notice.

For example : E.division_polynomial(k) returned 4*x^3 + 4*A*x + 4*B but theory says it is 2y. From this one can get a intuition that may be we are squaring them.

But E.division_polynomial(8) returned a degree $33$ polynomial which means that clearly we are not squaring things.

order of a polynomial $f(x)$ in $\mathbb{F}_p [x]$ is defined as minimum $e$ such that $f(x) | x^e -1$ . Do we have an inbuilt function in sage to find the same ?

For any matrix A when we type

A.kernel()

It returns

 A.kernel()
Free module of degree 45 and rank 12 over Integer Ring
Echelon basis matrix:`[ 1  0 -1][ 0  1  2][ 0  0  0]`

How to I access this Echelon basis matrix directly?

I have tried this

A.kernel.echelon_form()

but it says

TypeError: echelon_form() takes at least 2 arguments (1 given)

When I tried to find it directly using

A.kernel()

it said

 Cannot compute a matrix kernel over Ring of integers modulo 11053185041

Thanks a lot ! Can you kindly help me with this question also.

Consider the following code

Zn=Zmod(n)
R = PolynomialRing(Zn,'x')
F = R.quotient((x**r)-1)
y=F((x+1))
f=F(y**n)

Clearly f will be a polynomial in xbar , I want to consider this polynomial as a polynomial in $ \mathbb{Z}[x] $ and evaluate its discriminant.

I tried "f.polynomial()" but it is not working. Any suggestions ?

@tmonteil In your example Cyclotomic polynomial factors (here it splits that is has roots in GF(p) ), but what if it is irreducible (which it would be when ord_r(p)=r-1, take r=5 in your example). In that case your algorithms gives "No roots". I am interested in the cases when y doesn't lie in GF(p) but in some extension of it.

One can choose 'a' to be any quadratic residue modulo p

I want to compute $$ gcd_{X}((X-y)^2 -a , X^{\frac{q-1}{2}}-1)$$ with respect to X(taking y as a field constant).

I can't see any direct implementation of this in sage. Can any one suggest how to implement it.

Here Arithmetic is over $GF(p)$ and y is root of cyclotomic polynomial of degree r over $GF(p)$ and $q = p^r$

Consider the following code :-

p=104711
r=101
K=GF(p)
L.<y>=K.extension(cyclotomic_polynomial(r))
for a in range(10000,10000000):
    f=(y+1)^(a)
    L=f.list()
    if L.count(0)>1:
        print "special low support %d" %a
        for i in range(100):
            if L[i]==0:
                print i

I want the list of coefficient of f(y) but when L is a field (which it for the above stated parameters ) it is giving this error.

AttributeError:
'sage.rings.finite_rings.element_pari_ffelt.FiniteFieldElement_pari_ffel\
t' object has no attribute 'list'

when L is a ring it is working perfectly. Also when I try to use f.coeff() it basically consider f(y) as a constant.