Here is a system of polynomials (posted on sage-support forum) which
maxima can't solve:
(%i1) [x^3 + 3*x^2*y + 3*x*y^2 = 45487, y^3 =
549755813888];
(%o1) [3*x*y^2+3*x^2*y+x^3=45487,y^3=549755813888]
(%i2) solve(%);
allroots: no roots found.
-- an error. To debug this try: debugmode(true);
The problem is in allroots called from alysys. I patched it so that I
can force it to use the algorithm for complex polynomials. With this
the system can be solved:
(%i3) solve(%o1), allroots_complex=true;
(%o3) [[y=7094.480107802121*%i-4095.999999999999,x=1.9566663332522818*10^-4*%i-1.1296818342174827*10^-4],[y=-7094.480107802121*%i-4096.000000000002,x=-1.9566663332522818*10^-4*%i-1.1296818342174833*10^-4],[y=7094.480107802121*%i-4095.999999999999,x=-14188.96041127087*%i-1.1296818443896424*10^-4],[y=7094.480107802121*%i-4095.999999999999,x=12288.00022593637-7094.480107802124*%i],[y=-7094.480107802121*%i-4096.000000000002,x=14188.96041127087*%i-1.1296818335232472*10^-4],[y=-7094.480107802121*%i-4096.000000000002,x=7094.480107802125*%i+12288.00022593637],[y=8192.0,x=2.2593636684349655*10^-4-2.710505431213761*10^-20*%i],[y=8192.0,x=7094.480303463693*%i-12288.00011296539],[y=8192.0,x=-7094.480303463693*%i-12288.00011297098]]
Maybe something like this should be available in Maxima?
Andrej