Getting nowhere with maple using its triangularize and groebner decompositions for even moderate size systems with any symbolic factors. Any suggestions on how better to approach this would be appreciated. To put a shape on this query I append a sample problem case below, u,v,w,x,y,z are variables, a,b,c,d constants. 


6-cvu+cu+cv-2uv+8xy+8wy+8xz+8wz+8yz+8wx-8yzuv+16zyxw-c


-6x+2xu-6w+2wv-6y+2yuv-8yxw-6z+2zu+2zv+2zuv-8zxw-8zyw+8zywv-8zyx+8zyxu+2yu+2yv


4-4xyu-4wyv-4wzv-4xzu-4yzu-4yzv-uvd+ud+vd+4xy+4wy+4xz+4wz+4yz+4wx-2v-2u+4yzuv-d


-2w+2wv-2x+2xu-2y+2yu+2yv-2yuv-2z+2zu+2zv-2zuv


1+u+v+uv-a+ua+va-uva


4+4xyu+4wyv+4wzv+4xzu+4yzu+4yzv-uvb+ub+vb+4xy+4wy+4xz+4wz+4yz+4wx+2v+2u+4yzuv-b



Reducing system of polynomials with symbolic factors
**Added more details:** From specializing $a,b,c,d$ to random integers it seems that your system is $0$-dimensional, and $z$ is the root of a degree $16$ polynomial $f\_{a,b,c,d}(z)$. You might compute these degree $16$ polynomials for many integer tuples $a,b,c,d$, and try to guess/interpolate the dependencies of the coefficients of $f\_{a,b,c,d}(z)$ on $a,b,c,d$. Once you know $f\_{a,b,c,d}(z)$, the rest of the computation should be cheap.


If one fixes all but one of the parameters $a,b,c,d$, and lets the remaining one run through some possibilities, one can guess the degrees of the dependencies of the coefficients of $f\_{a,b,c,d}$ in $a,b,c,d$, and then solve the corresponding interpolation problem. For instance, the coefficient of $z^{14}$ seems to be $-3a/4+d-13/4$. The highest degree dependency seems to occur for the coefficient of $z^4$, of degrees $5,3,3,4$ in $a,b,c,d$, respectively. So in order to compute this coefficient, one has to compute $6\cdot 4\cdot 4\cdot 5=480$ examples, and solve the corresponding linear system of equations in $480$ unknowns. Don't know if that is possible, but I would expect it can be done.


The Sage code for the computation of $f\_{a,b,c,d}$ is:



```
while True:
    a,b,c,d=(floor(20*random()-10) for i in range(4))
    R.<u,v,w,x,y,z>=PolynomialRing(QQ,6,order='lex')
    l=[6-c*v*u+ ... SNIP!!! ... +4*y*z*u*v-b]
    i=ideal(l)
    g=i.groebner_basis()
    zz=g[5]
    print (a,b,c,d),zz/zz.coefficient({z:16})

```

produces



```

(9, 4, 9, 5) z^16 - 5*z^14 + 47/4*z^12 - 833/64*z^10 + 2263/256*z^8 + 3973/1024*z^6 - 27697/4096*z^4 + 5775/4096*z^2 + 625/4096
(-7, 7, 5, -1) z^16 + z^14 + 13/8*z^12 - 23/8*z^10 + 3011/256*z^8 - 1193/128*z^6 + 16581/2048*z^4 + 2303/4096*z^2 + 2401/65536
(2, -7, 3, 1) z^16 - 15/4*z^14 + 29/4*z^12 - 305/32*z^10 + 75/8*z^8 - 1703/256*z^6 + 453/128*z^4 - 1233/1024*z^2 + 81/256
(-2, -5, -10, 2) z^16 + 1/4*z^14 - 17/8*z^12 - 41/32*z^10 + 457/256*z^8 + 1133/1024*z^6 - 605/1024*z^4 - 21/64*z^2 + 1/16
(-10, -3, -9, 4) z^16 + 33/4*z^14 + 219/8*z^12 + 1023/64*z^10 + 4017/256*z^8 - 6919/1024*z^6 - 1301/1024*z^4 - 138985/16384*z^2 + 130321/65536
(-4, 5, -4, 7) z^16 + 27/4*z^14 + 61/4*z^12 + 79/64*z^10 - 1293/64*z^8 + 4333/1024*z^6 + 24929/2048*z^4 - 40071/4096*z^2 + 130321/65536
(-7, -6, -9, 0) z^16 + 2*z^14 + z^12 - 23/4*z^10 + 643/128*z^8 - 331/128*z^6 + 171/256*z^4 - 243/256*z^2 + 6561/65536
... snip ...

```

