36 KiB
36 KiB
Theory
Let $C : y^m = f(x)$. Then:
the basis of $H^0(C, \Omega_{C/k})$ is given by: $$x^{i-1} dx/y^j,$$ where $1 \le i \le r-1$, $1 \le j \le m-1$, $-mi + rj \ge \delta$ and $\delta := GCD(m, r)$, $r := \deg f$.
# The program computes the basis of holomorphic differentials of y^m = f(x) in char p.
# The coefficient j means that we compute the j-th eigenpart, i.e.
# forms y^j * f(x) dx. Output is [f(x), 0]
def baza_holo(m, f, j, p):
R.<x> = PolynomialRing(GF(p))
f = R(f)
r = f.degree()
delta = GCD(m, r)
baza = {}
k = 0
for i in range(1, r):
if (r*j - m*i >= delta):
baza[k] = [x^(i-1), R(0)]
k = k+1
return baza# The program computes the basis of de Rham cohomology of y^m = f(x) in char p.
# We treat them as pairs [omega, f], where omega is regular on the affine part
# and omega - df is regular on the second atlas.
# The coefficient j means that we compute the j-th eigenpart, i.e.
# [y^j * f(x) dx, g(x)/y^j]. Output is [f(x), g(x)]
def baza_dr(m, f, j, p):
R.<x> = PolynomialRing(GF(p))
f = R(f)
r = f.degree()
delta = GCD(m, r)
baza = {}
holo = baza_holo(m, f, j, p)
for k in range(0, len(holo)):
baza[k] = holo[k]
k = len(baza)
for i in range(1, r):
if (r*(m-j) - m*i >= delta):
s = R(m-j)*R(x)*R(f.derivative()) - R(m)*R(i)*f
psi = R(obciecie(s, i, p))
baza[k] = [psi, R(m)/x^i]
k = k+1
return baza#auxiliary programs
def stopnie_bazy_holo(m, f, j, p):
baza = baza_holo(m, f, j, p)
stopnie = {}
for k in range(0, len(baza)):
stopnie[k] = baza[k][0].degree()
return stopnie
def stopnie_bazy_dr(m, f, j, p):
baza = baza_dr(m, f, j, p)
stopnie = {}
for k in range(0, len(baza)):
stopnie[k] = baza[k][0].degree()
return stopnie
def stopnie_drugiej_wspolrzednej_bazy_dr(m, f, j, p):
baza = baza_dr(m, f, j, p)
stopnie = {}
for k in range(0, len(baza)):
if baza[k][1] != 0:
stopnie[k] = baza[k][1].denominator().degree()
return stopnie
def obciecie(f, i, p):
R.<x> = PolynomialRing(GF(p))
f = R(f)
coeff = f.coefficients(sparse = false)
return sum(x^(j-i-1) * coeff[j] for j in range(i+1, f.degree() + 1))
#Any element [f dx, g] is represented as a combination of the basis vectors.
def zapis_w_bazie_dr(elt, m, f, j, p):
#print(elt)
R.<x> = PolynomialRing(GF(p))
RR = FractionField(R)
f = R(f)
r = f.degree()
delta = GCD(m, r)
baza = baza_dr(m, f, j, p)
wymiar = len(baza)
zapis = vector([GF(p)(0) for i in baza])
stopnie = stopnie_bazy_dr(m, f, j, p)
inv_stopnie = {v: k for k, v in stopnie.items()}
stopnie_holo = stopnie_bazy_holo(m, f, j, p)
inv_stopnie_holo = {v: k for k, v in stopnie_holo.items()}
## zmiana
if elt[0]== 0 and elt[1] == 0:
print('p1')
return zapis
if elt[1] == 0:
print('p2')
elt[0] = R(elt[0])
d = elt[0].degree()
a = elt[0].coefficients(sparse = false)[d]
k = inv_stopnie_holo[d] #ktory element bazy jest stopnia d? ten o indeksie k
a1 = baza[k][0].coefficients(sparse = false)[d]
elt1 = [R(0),R(0)]
elt1[0] = elt[0] - a/a1 * baza[k][0]
elt1[1] = R(0)
return zapis_w_bazie_dr(elt1, m, f, j, p) + vector([a/a1*GF(p)(i == k) for i in range(0, len(baza))])
g = elt[1]
g1 = R(elt[1].numerator())
g2 = R(elt[1].denominator())
d1 = g1.degree()
d2 = g2.degree()
a1 = g1.coefficients(sparse = false)[d1]
a2 = g2.coefficients(sparse = false)[d2]
a = a1/a2
d = d2 - d1
Rr = r/delta
M = m/delta
stopnie2 = stopnie_drugiej_wspolrzednej_bazy_dr(m, f, j, p)
inv_stopnie2 = {v: k for k, v in stopnie2.items()}
if (d not in inv_stopnie2):
print('p3')
if d>= 0:
elt1 = [elt[0], RR(elt[1]) - a1/a2*1/R(x^d)]
else:
elt1 = [elt[0], RR(elt[1]) - a1/a2*R(x^(-d))]
return zapis_w_bazie_dr(elt1, m, f, j, p)
print('p4')
k = inv_stopnie2[d]
elt1 = [R(0), R(0)]
elt1[0] = elt[0] - a*baza[k][0]
elt1[1] = elt[1] - a*baza[k][1]
return zapis_w_bazie_dr(elt1, m, f, j, p) + vector([a*GF(p)(i == k) for i in range(0, len(baza))])
def zapis_w_bazie_holo(elt, m, f, j, p):
R.<x> = PolynomialRing(GF(p))
f = R(f)
r = f.degree()
delta = GCD(m, r)
baza = baza_holo(m, f, j, p)
wymiar = len(baza)
zapis = vector([GF(p)(0) for i in baza])
stopnie = stopnie_bazy_holo(m, f, j, p)
inv_stopnie = {v: k for k, v in stopnie.items()}
if elt[0] == 0:
return zapis
d = elt[0].degree()
a = elt[0].coefficients(sparse = false)[d]
k = inv_stopnie[d] #ktory element bazy jest stopnia d? ten o indeksie k
a1 = baza[k][0].coefficients(sparse = false)[d]
elt1 = [R(0),R(0)]
elt1[0] = elt[0] - a/a1 * baza[k][0]
return zapis_w_bazie_holo(elt1, m, f, j, p) + vector([a/a1 * GF(p)(i == k) for i in range(0, len(baza))])
We have: $V(\omega, f) = (C(\omega), 0)$ and $F(\omega, f) = (0, f^p)$, where C denotes the Cartier operator. Moreover:
let $t = multord_m(p)$, $M := (p^t - 1)/m$. Then: $y^{p^t - 1} = f(x)^M$ and $1/y = f(x)^M/y^{p^t}$. Thus:
$$ C(P(x) , dx / y^j) = C(P(x) , f(x)^{M \cdot j} , dx /y^{p^t \cdot j}) = \frac{1}{y^{p^{t - 1} \cdot j}} C(P(x) , f(x)^{M \cdot j} , dx) = \frac{1}{y^{(p^{t - 1} \cdot j) , mod , m}} \cdot \frac{1}{f(x)^{[p^{t - 1} \cdot j/m]}} \cdot C(P(x) , f(x)^{M \cdot j} , dx)$$
def czesc_wielomianu(p, h):
R.<x> = PolynomialRing(GF(p))
h = R(h)
wynik = R(0)
for i in range(0, h.degree()+1):
if (i%p) == p-1:
potega = Integer((i-(p-1))/p)
wynik = wynik + Integer(h[i]) * x^(potega)
return wynik
def cartier_dr(p, m, f, elt, j): #Cartier na y^m = f dla elt = [forma rozniczkowa, fkcja]
R.<x> = PolynomialRing(GF(p))
f = R(f)
r = f.degree()
delta = GCD(m, r)
rzad = Integers(m)(p).multiplicative_order()
M = Integer((p^(rzad)-1)/m)
W = R(elt[0])
h = R(W*f^(M*j))
B = floor(p^(rzad-1)*j/m)
g = czesc_wielomianu(p, h)/f^B
jj = (p^(rzad-1)*j)%m
#jj = Integers(m)(j/p)
return [g, 0] #jest to w czesci indeksowanej jj
def macierz_cartier_dr(p, m, f, j):
baza = baza_dr(m, f, j, p)
A = matrix(GF(p), len(baza), len(baza))
for k in range(0, len(baza)):
cart = cartier_dr(p, m, f, baza[k], j)
v = zapis_w_bazie_dr(cart, m, f, j, p)
A[k, :] = matrix(v)
return A.transpose()$F((\omega, P(x) \cdot y^j)) = (0, P(x)^p \cdot y^{p \cdot j}) = (0, P(x)^p \cdot f(x)^{[p \cdot j/m]} \cdot y^{(p \cdot j) , mod , m})$
def frobenius_dr(p, m, f, elt, j): #Cartier na y^m = f dla elt = [forma rozniczkowa, fkcja]
R.<x> = PolynomialRing(GF(p))
RR = FractionField(R)
f = R(f)
M = floor(j*p/m)
return [0, f^M * RR(elt[1])^p] #eigenspace = j*p mod m
def macierz_frob_dr(p, m, f, j):
baza = baza_dr(m, f, j, p)
A = matrix(GF(p), len(baza), len(baza))
for k in range(0, len(baza)):
frob = frobenius_dr(p, m, f, baza[k], j)
v = zapis_w_bazie_dr(frob, m, f, j, p)
A[k, :] = matrix(v)
return A.transpose()m = 2
j = 1
p = 5
f = x^3 + x+3
baza_dr(m, f, j, p)
#macierz_frob_dr(p, m, f, j){0: [1, 0], 1: [x, 2/x]}frobenius_dr(p, m, f, [x, 2/x], j)[0, (2*x^6 + 4*x^4 + 2*x^3 + 2*x^2 + 2*x + 3)/x^5]
zapis_w_bazie_dr([0, (2*x^6 + 4*x^4 + 2*x^3 + 2*x^2 + 2*x + 3)/x^5], m, f, j, p)p3 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4 p4
[0;31m---------------------------------------------------------------------------[0m [0;31mRecursionError[0m Traceback (most recent call last) [0;32m<ipython-input-160-b5994066838e>[0m in [0;36m<module>[0;34m()[0m [0;32m----> 1[0;31m [0mzapis_w_bazie_dr[0m[0;34m([0m[0;34m[[0m[0mInteger[0m[0;34m([0m[0;36m0[0m[0;34m)[0m[0;34m,[0m [0;34m([0m[0mInteger[0m[0;34m([0m[0;36m2[0m[0;34m)[0m[0;34m*[0m[0mx[0m[0;34m**[0m[0mInteger[0m[0;34m([0m[0;36m6[0m[0;34m)[0m [0;34m+[0m [0mInteger[0m[0;34m([0m[0;36m4[0m[0;34m)[0m[0;34m*[0m[0mx[0m[0;34m**[0m[0mInteger[0m[0;34m([0m[0;36m4[0m[0;34m)[0m [0;34m+[0m [0mInteger[0m[0;34m([0m[0;36m2[0m[0;34m)[0m[0;34m*[0m[0mx[0m[0;34m**[0m[0mInteger[0m[0;34m([0m[0;36m3[0m[0;34m)[0m [0;34m+[0m [0mInteger[0m[0;34m([0m[0;36m2[0m[0;34m)[0m[0;34m*[0m[0mx[0m[0;34m**[0m[0mInteger[0m[0;34m([0m[0;36m2[0m[0;34m)[0m [0;34m+[0m [0mInteger[0m[0;34m([0m[0;36m2[0m[0;34m)[0m[0;34m*[0m[0mx[0m [0;34m+[0m [0mInteger[0m[0;34m([0m[0;36m3[0m[0;34m)[0m[0;34m)[0m[0;34m/[0m[0mx[0m[0;34m**[0m[0mInteger[0m[0;34m([0m[0;36m5[0m[0;34m)[0m[0;34m][0m[0;34m,[0m [0mm[0m[0;34m,[0m [0mf[0m[0;34m,[0m [0mj[0m[0;34m,[0m [0mp[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m [0;32m<ipython-input-155-baa93df16142>[0m in [0;36mzapis_w_bazie_dr[0;34m(elt, m, f, j, p)[0m [1;32m 85[0m [0;32melse[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 86[0m [0melt1[0m [0;34m=[0m [0;34m[[0m[0melt[0m[0;34m[[0m[0mInteger[0m[0;34m([0m[0;36m0[0m[0;34m)[0m[0;34m][0m[0;34m,[0m [0mRR[0m[0;34m([0m[0melt[0m[0;34m[[0m[0mInteger[0m[0;34m([0m[0;36m1[0m[0;34m)[0m[0;34m][0m[0;34m)[0m [0;34m-[0m [0ma1[0m[0;34m/[0m[0ma2[0m[0;34m*[0m[0mR[0m[0;34m([0m[0mx[0m[0;34m**[0m[0;34m([0m[0;34m-[0m[0md[0m[0;34m)[0m[0;34m)[0m[0;34m][0m[0;34m[0m[0;34m[0m[0m [0;32m---> 87[0;31m [0;32mreturn[0m [0mzapis_w_bazie_dr[0m[0;34m([0m[0melt1[0m[0;34m,[0m [0mm[0m[0;34m,[0m [0mf[0m[0;34m,[0m [0mj[0m[0;34m,[0m [0mp[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 88[0m [0;34m[0m[0m [1;32m 89[0m [0mprint[0m[0;34m([0m[0;34m'p4'[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m<ipython-input-155-baa93df16142>[0m in [0;36mzapis_w_bazie_dr[0;34m(elt, m, f, j, p)[0m [1;32m 92[0m [0melt1[0m[0;34m[[0m[0mInteger[0m[0;34m([0m[0;36m0[0m[0;34m)[0m[0;34m][0m [0;34m=[0m [0melt[0m[0;34m[[0m[0mInteger[0m[0;34m([0m[0;36m0[0m[0;34m)[0m[0;34m][0m [0;34m-[0m [0ma[0m[0;34m*[0m[0mbaza[0m[0;34m[[0m[0mk[0m[0;34m][0m[0;34m[[0m[0mInteger[0m[0;34m([0m[0;36m0[0m[0;34m)[0m[0;34m][0m[0;34m[0m[0;34m[0m[0m [1;32m 93[0m [0melt1[0m[0;34m[[0m[0mInteger[0m[0;34m([0m[0;36m1[0m[0;34m)[0m[0;34m][0m [0;34m=[0m [0melt[0m[0;34m[[0m[0mInteger[0m[0;34m([0m[0;36m1[0m[0;34m)[0m[0;34m][0m [0;34m-[0m [0ma[0m[0;34m*[0m[0mbaza[0m[0;34m[[0m[0mk[0m[0;34m][0m[0;34m[[0m[0mInteger[0m[0;34m([0m[0;36m1[0m[0;34m)[0m[0;34m][0m[0;34m[0m[0;34m[0m[0m [0;32m---> 94[0;31m [0;32mreturn[0m [0mzapis_w_bazie_dr[0m[0;34m([0m[0melt1[0m[0;34m,[0m [0mm[0m[0;34m,[0m [0mf[0m[0;34m,[0m [0mj[0m[0;34m,[0m [0mp[0m[0;34m)[0m [0;34m+[0m [0mvector[0m[0;34m([0m[0;34m[[0m[0ma[0m[0;34m*[0m[0mGF[0m[0;34m([0m[0mp[0m[0;34m)[0m[0;34m([0m[0mi[0m [0;34m==[0m [0mk[0m[0;34m)[0m [0;32mfor[0m [0mi[0m [0;32min[0m [0mrange[0m[0;34m([0m[0mInteger[0m[0;34m([0m[0;36m0[0m[0;34m)[0m[0;34m,[0m [0mlen[0m[0;34m([0m[0mbaza[0m[0;34m)[0m[0;34m)[0m[0;34m][0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 95[0m [0;34m[0m[0m [1;32m 96[0m [0;34m[0m[0m ... last 1 frames repeated, from the frame below ... [0;32m<ipython-input-155-baa93df16142>[0m in [0;36mzapis_w_bazie_dr[0;34m(elt, m, f, j, p)[0m [1;32m 92[0m [0melt1[0m[0;34m[[0m[0mInteger[0m[0;34m([0m[0;36m0[0m[0;34m)[0m[0;34m][0m [0;34m=[0m [0melt[0m[0;34m[[0m[0mInteger[0m[0;34m([0m[0;36m0[0m[0;34m)[0m[0;34m][0m [0;34m-[0m [0ma[0m[0;34m*[0m[0mbaza[0m[0;34m[[0m[0mk[0m[0;34m][0m[0;34m[[0m[0mInteger[0m[0;34m([0m[0;36m0[0m[0;34m)[0m[0;34m][0m[0;34m[0m[0;34m[0m[0m [1;32m 93[0m [0melt1[0m[0;34m[[0m[0mInteger[0m[0;34m([0m[0;36m1[0m[0;34m)[0m[0;34m][0m [0;34m=[0m [0melt[0m[0;34m[[0m[0mInteger[0m[0;34m([0m[0;36m1[0m[0;34m)[0m[0;34m][0m [0;34m-[0m [0ma[0m[0;34m*[0m[0mbaza[0m[0;34m[[0m[0mk[0m[0;34m][0m[0;34m[[0m[0mInteger[0m[0;34m([0m[0;36m1[0m[0;34m)[0m[0;34m][0m[0;34m[0m[0;34m[0m[0m [0;32m---> 94[0;31m [0;32mreturn[0m [0mzapis_w_bazie_dr[0m[0;34m([0m[0melt1[0m[0;34m,[0m [0mm[0m[0;34m,[0m [0mf[0m[0;34m,[0m [0mj[0m[0;34m,[0m [0mp[0m[0;34m)[0m [0;34m+[0m [0mvector[0m[0;34m([0m[0;34m[[0m[0ma[0m[0;34m*[0m[0mGF[0m[0;34m([0m[0mp[0m[0;34m)[0m[0;34m([0m[0mi[0m [0;34m==[0m [0mk[0m[0;34m)[0m [0;32mfor[0m [0mi[0m [0;32min[0m [0mrange[0m[0;34m([0m[0mInteger[0m[0;34m([0m[0;36m0[0m[0;34m)[0m[0;34m,[0m [0mlen[0m[0;34m([0m[0mbaza[0m[0;34m)[0m[0;34m)[0m[0;34m][0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 95[0m [0;34m[0m[0m [1;32m 96[0m [0;34m[0m[0m [0;31mRecursionError[0m: maximum recursion depth exceeded while calling a Python object
h = sum(i*x^i for i in range(0, p^2))
czesc_wielomianu(p, h)4*x^4 + 4*x^3 + 4*x^2 + 4*x + 4
m = 5
j = 1
p = 5
f = x^4 + x+2
baza_dr(m, f, j, p){0: [x^2, 0], 1: [x, 0], 2: [1, 0]}macierz_frob_dr(p, m, f, j)[0 0 0] [0 0 0] [0 0 0]