3D rendering not yet implemented
[3375*a + 729*b + 2025*d + 1215*f, 1800*e + 648*g + 1080*j, 10125*a - 2187*b + 2025*d - 1215*f + 960*h + 576*i, 512*c + 4050*e - 1134*g + 270*j, 10125*a + 2187*b - 2025*d - 1215*f + 1440*h - 288*i, 384*c + 2700*e + 324*g - 1080*j, 3375*a - 729*b - 2025*d + 1215*f + 540*h - 252*i, 96*c + 450*e + 162*g - 270*j, 60*h - 36*i, 8*c]
[[a == r1, b == -5/3*r1, c == 0, d == 1/3*r1, e == 0, f == -7/3*r1, g == 0, h == -9*r1, i == -15*r1, j == 0]]
(-6912, -221184)
0
+Infinity
[x == (1/2), x == -1, x == (-1/4)]
(-27648, -1769472)
0
+Infinity
(-1, 0)
-4
1728
[[x == 1, y == 0]]
[x == -2, x == -1]
4*a^4*b^3*c^2*d - 4*a^3*b^4*c^2*d - 4*a^4*b^2*c^3*d + 4*a^2*b^4*c^3*d + 4*a^3*b^2*c^4*d - 4*a^2*b^3*c^4*d - 4*a^4*b^3*c*d^2 + 4*a^3*b^4*c*d^2 + 4*a^4*b*c^3*d^2 - 4*a*b^4*c^3*d^2 - 4*a^3*b*c^4*d^2 + 4*a*b^3*c^4*d^2 + 4*a^4*b^2*c*d^3 - 4*a^2*b^4*c*d^3 - 4*a^4*b*c^2*d^3 + 4*a*b^4*c^2*d^3 + 4*a^2*b*c^4*d^3 - 4*a*b^2*c^4*d^3 - 4*a^3*b^2*c*d^4 + 4*a^2*b^3*c*d^4 + 4*a^3*b*c^2*d^4 - 4*a*b^3*c^2*d^4 - 4*a^2*b*c^3*d^4 + 4*a*b^2*c^3*d^4 - a^4*b^3*c + a^3*b^4*c + a^4*b*c^3 - a*b^4*c^3 - a^3*b*c^4 + a*b^3*c^4 + a^4*b^3*d - a^3*b^4*d - a^4*c^3*d + b^4*c^3*d + a^3*c^4*d - b^3*c^4*d - a^4*b*d^3 + a*b^4*d^3 + a^4*c*d^3 - b^4*c*d^3 - a*c^4*d^3 + b*c^4*d^3 + a^3*b*d^4 - a*b^3*d^4 - a^3*c*d^4 + b^3*c*d^4 + a*c^3*d^4 - b*c^3*d^4 + 4*a^3*b^2*c - 4*a^2*b^3*c - 4*a^3*b*c^2 + 4*a*b^3*c^2 + 4*a^2*b*c^3 - 4*a*b^2*c^3 - 4*a^3*b^2*d + 4*a^2*b^3*d + 4*a^3*c^2*d - 4*b^3*c^2*d - 4*a^2*c^3*d + 4*b^2*c^3*d + 4*a^3*b*d^2 - 4*a*b^3*d^2 - 4*a^3*c*d^2 + 4*b^3*c*d^2 + 4*a*c^3*d^2 - 4*b*c^3*d^2 - 4*a^2*b*d^3 + 4*a*b^2*d^3 + 4*a^2*c*d^3 - 4*b^2*c*d^3 - 4*a*c^2*d^3 + 4*b*c^2*d^3
(4*a*b*c*d - a*b - a*c - b*c - a*d - b*d - c*d + 4)*(a - b)*(a - c)*(a - d)*(b - c)*(b - d)*(c - d)
4*a*b*c*d - a*b - a*c - b*c - a*d - b*d - c*d + 4
\left(\right)
The symmetric expression F is
Solve F=0 for t4 and set it equal to f
-\frac{q t_{1} + {\left(r t_{1} + q\right)} t_{2} + {\left(r t_{1} + {\left(s t_{1} + r\right)} t_{2} + q\right)} t_{3} + p}{r t_{1} + {\left(s t_{1} + r\right)} t_{2} + {\left(s t_{1} + {\left(s + t_{1}\right)} t_{2} + r\right)} t_{3} + q}
Quotient of the partial derivatives of f with respect to t2 and t3:
The above expression should be independent of t1, so take the partial derivative wrt t1, take the numerator of that and factorize:
Since t1, t2, t3 are arbitrary in some small ball, the first and last factors are non-zero generically, and we obtain that the catalecticant vanishes.
s^{2} t_{1}^{2} t_{2}^{2} + r s t_{1}^{2} t_{2} + r s t_{1} t_{2}^{2} - r t_{1}^{2} t_{2}^{2} + r^{2} t_{1}^{2} - q s t_{1}^{2} + r^{2} t_{1} t_{2} - q t_{1}^{2} t_{2} + r^{2} t_{2}^{2} - q s t_{2}^{2} - q t_{1} t_{2}^{2} + q r t_{1} - p s t_{1} + q r t_{2} - p s t_{2} - p t_{1} t_{2} + q^{2} - p r
-3/2*a*b + a*c + b*c - c^2 + 1/2*a*d + 1/2*b*d - 1/2*d^2
Ideal (-3/2*a*b + a*c + b*c - c^2 + 1/2*a*d + 1/2*b*d - 1/2*d^2) of Multivariate Polynomial Ring in a, b, c, d over Rational Field
a^3*b^3 + a^3*b^2*c + a^2*b^3*c + a^3*b*c^2 + a^2*b^2*c^2 + a*b^3*c^2 + a^3*c^3 + a^2*b*c^3 + a*b^2*c^3 + b^3*c^3 + a^3*b^2*d + a^2*b^3*d + a^3*b*c*d + 2*a^2*b^2*c*d + a*b^3*c*d + a^3*c^2*d + 2*a^2*b*c^2*d + 2*a*b^2*c^2*d + b^3*c^2*d + a^2*c^3*d + a*b*c^3*d + b^2*c^3*d + a^3*b*d^2 + a^2*b^2*d^2 + a*b^3*d^2 + a^3*c*d^2 + 2*a^2*b*c*d^2 + 2*a*b^2*c*d^2 + b^3*c*d^2 + a^2*c^2*d^2 + 2*a*b*c^2*d^2 + b^2*c^2*d^2 + a*c^3*d^2 + b*c^3*d^2 + a^3*d^3 + a^2*b*d^3 + a*b^2*d^3 + b^3*d^3 + a^2*c*d^3 + a*b*c*d^3 + b^2*c*d^3 + a*c^2*d^3 + b*c^2*d^3 + c^3*d^3
False
(-1/81) * (-195*a^3*c^3 - 195*b^3*c^3 - 48*a^2*c^4 - 48*b^2*c^4 + 38*a*c^5 + 38*b*c^5 + 124*c^6 - 270*a^3*c^2*d - 270*b^3*c^2*d - 438*a^2*c^3*d - 438*b^2*c^3*d - 184*a*c^4*d - 184*b*c^4*d + 530*c^5*d - 225*a^3*c*d^2 - 225*b^3*c*d^2 - 468*a^2*c^2*d^2 - 468*b^2*c^2*d^2 - 649*a*c^3*d^2 - 649*b*c^3*d^2 + 764*c^4*d^2 - 120*a^3*d^3 - 120*b^3*d^3 - 258*a^2*c*d^3 - 258*b^2*c*d^3 - 434*a*c^2*d^3 - 434*b*c^2*d^3 + 474*c^3*d^3 - 3*a^2*d^4 - 3*b^2*d^4 - 14*a*c*d^4 - 14*b*c*d^4 + 379*c^2*d^4 + 28*a*d^5 + 28*b*d^5 + 145*c*d^5 + 14*d^6)
Error in lines 11-11
Traceback (most recent call last):
File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1013, in execute
exec compile(block+'\n', '', 'single') in namespace, locals
File "", line 1, in <module>
File "sage/structure/element.pyx", line 484, in sage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4377)
return self.getattr_from_category(name)
File "sage/structure/element.pyx", line 497, in sage.structure.element.Element.getattr_from_category (build/cythonized/sage/structure/element.c:4486)
return getattr_from_other_class(self, cls, name)
File "sage/cpython/getattr.pyx", line 254, in sage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:1901)
raise dummy_attribute_error
AttributeError: 'sage.rings.fraction_field_element.FractionFieldElement' object has no attribute 'simplify'
\left(\right)
\left(\right)
\left(\right)
\left(\right)
-4*(r^3 - 2*q*r*s + p*s^2 - q^2 + p*r)*(q^2 - p*r)
-2*(r^3 - 2*q*r*s + p*s^2 - q^2 + p*r)*(q*r - p*s)
-4*(r^3 - 2*q*r*s + p*s^2 - q^2 + p*r)*(q*s + p)
2*(r^3 - 2*q*r*s + p*s^2 - q^2 + p*r)*(r^2 - 2*q*s - p)
-2*(r^3 - 2*q*r*s + p*s^2 - q^2 + p*r)*(r*s + q)
-2*(r^3 - 2*q*r*s + p*s^2 - q^2 + p*r)*(s^2 + r)
\left(\right)
[x == -1/4*(4*r*s^2 - 5*r^2 + p)/(2*s^3 - 3*r*s + q)]
[, , , ]
4
0