# Revision history [back]

### How to compute common zeros of system of polynomial equations with dimension 2?

I have some ideal of homogenous polynomials defined over some finite field: J is the ideal of interest. However I can't call J.variety() since it is not zero dimensional. The system of equations might contain 6 or 231 polynomials in four variables:

  sage: [I.gens() for I in J.minimal_associated_primes()]
[[x1 + x2, x0 + x3], [x2 + x3, x0 + x1], [x1 + x3, x0 + x2]]
sage: J.dimension()
2


Any ideas?

### How to compute common zeros of system of polynomial equations with dimension 2?

I have some ideal of homogenous polynomials defined over some finite field: J is the ideal of interest. However I can't call J.variety() since it is not zero dimensional. The system of equations might contain 6 or 231 polynomials in four variables:

  sage: [I.gens() for I in J.minimal_associated_primes()]
[[x1 + x2, x0 + x3], [x2 + x3, x0 + x1], [x1 + x3, x0 + x2]]
sage: J.dimension()
2


Any ideas?

On request. The program computes the invariant under the group $2_{+}^{1+2\cdot2}$ homogenous polynomials of given degree. To construct the polynomials and get a list of them call: homInvar(6):

F = ZZ;
a = matrix(F, [[0,0,0,-1],[0,0,1,0],[0,-1,0,0],[1,0,0,0]])
b = matrix(F, [[1,0,0,0],[0,1,0,0],[0,0,-1,0],[0,0,0,-1]])
c = matrix(F, [[0,1,0,0],[-1,0,0,0],[0,0,0,1],[0,0,-1,0]])
d = matrix(F, [[0,1,0,0],[1,0,0,0],[0,0,0,-1],[0,0,-1,0]])

def getPart(deg):
part = []
for k in range(deg+1):
for l in range(deg+1):
for m in range(deg+1):
for n in range(deg+1):
if k+l+m+n == deg:
part.append((k,l,n,m))
return part

def p(x,part):
x = x.list()
pol = 0
for p in [part]:
mon = 1
for i in range(len(p)):
k = p[i]
mon = mon * x[i]**k
pol = pol + mon
return pol

def getGroup():
G = []
for k in range(4):
for l in range(4):
for m in range(4):
for n in range(4):
g = a**k*b**l*c**m*d**n
if G.count(g)==0:
G.append(g)
return G

def reynolds(Gr,part):
reyn = 0
n = len(part)
X = list(var('x%d' % i) for i in range(n))
x = matrix([X]).transpose()
for g in Gr:
reyn +=  p(g*x,part=part)
#print reyn
return 1/len(Gr)* reyn

def homInvar(deg,Gr=getGroup(),getPart=getPart):
parts = getPart(deg)
inv = set([])
for part in parts:
r = reynolds(Gr,part)
#print r
if r != 0:
return inv

 3 retagged FrédéricC 3517 ●3 ●34 ●69

### How to compute common zeros of system of polynomial equations with dimension 2?

I have some ideal of homogenous polynomials defined over some finite field: J is the ideal of interest. However I can't call J.variety() since it is not zero dimensional. The system of equations might contain 6 or 231 polynomials in four variables:

  sage: [I.gens() for I in J.minimal_associated_primes()]
[[x1 + x2, x0 + x3], [x2 + x3, x0 + x1], [x1 + x3, x0 + x2]]
sage: J.dimension()
2


Any ideas?

On request. The program computes the invariant under the group $2_{+}^{1+2\cdot2}$ homogenous polynomials of given degree. To construct the polynomials and get a list of them call: homInvar(6):

F = ZZ;
a = matrix(F, [[0,0,0,-1],[0,0,1,0],[0,-1,0,0],[1,0,0,0]])
b = matrix(F, [[1,0,0,0],[0,1,0,0],[0,0,-1,0],[0,0,0,-1]])
c = matrix(F, [[0,1,0,0],[-1,0,0,0],[0,0,0,1],[0,0,-1,0]])
d = matrix(F, [[0,1,0,0],[1,0,0,0],[0,0,0,-1],[0,0,-1,0]])

def getPart(deg):
part = []
for k in range(deg+1):
for l in range(deg+1):
for m in range(deg+1):
for n in range(deg+1):
if k+l+m+n == deg:
part.append((k,l,n,m))
return part

def p(x,part):
x = x.list()
pol = 0
for p in [part]:
mon = 1
for i in range(len(p)):
k = p[i]
mon = mon * x[i]**k
pol = pol + mon
return pol

def getGroup():
G = []
for k in range(4):
for l in range(4):
for m in range(4):
for n in range(4):
g = a**k*b**l*c**m*d**n
if G.count(g)==0:
G.append(g)
return G

def reynolds(Gr,part):
reyn = 0
n = len(part)
X = list(var('x%d' % i) for i in range(n))
x = matrix([X]).transpose()
for g in Gr:
reyn +=  p(g*x,part=part)
#print reyn
return 1/len(Gr)* reyn

def homInvar(deg,Gr=getGroup(),getPart=getPart):
parts = getPart(deg)
inv = set([])
for part in parts:
r = reynolds(Gr,part)
#print r
if r != 0: