# Reduce the number of equations

I have following 16 equations with 15 variables (XX is a variable, etc). I know there are only 6 independent variables. How can I reduce the number of equations below to get 9 independent equations?

2 x XX + 2 y YX + 2 z ZX == 2 X, 2 x XY + 2 y YY + 2 z ZY == 2 Y,
2 x XZ + 2 y YZ + 2 z ZZ == 2 Z, 2 X XX + 2 XY Y + 2 XZ Z == 2 x,
2 X YX + 2 Y YY + 2 YZ Z == 2 y, 2 X ZX + 2 Y ZY + 2 Z ZZ == 2 z,
2 x X + 2 YZ ZY - 2 YY ZZ == 2 XX, 2 x Y - 2 YZ ZX + 2 YX ZZ == 2 XY,
2 XZ == 2 x Z + 2 YY ZX - 2 YX ZY, 2 X y - 2 XZ ZY + 2 XY ZZ == 2 YX,
2 y Y + 2 XZ ZX - 2 XX ZZ == 2 YY, 2 YZ == 2 y Z - 2 XY ZX + 2 XX ZY,
2 XZ YY - 2 XY YZ + 2 X z == 2 ZX, -2 XZ YX + 2 XX YZ + 2 Y z == 2 ZY,
2 XY YX - 2 XX YY + 2 z Z == 2 ZZ, x^2 + X^2 + XX^2 + XY^2 + XZ^2 + y^2 + Y^2 + YX^2 + YY^2 + YZ^2 + z^2 + Z^2 + ZX^2 + ZY^2 + ZZ^2 == 3

• You could try Reduce Dec 14, 2021 at 15:15
• @UlrichNeumann I tried, but it ran forever...
– lol
Dec 14, 2021 at 17:42
• If you know which are the independent variables you could try Eliminate! Dec 14, 2021 at 17:43
• @UlrichNeumann I don't know which are independent variables. I only know how many...
– lol
Dec 14, 2021 at 17:44

You can find variables to take as parameters by computing a Groebner basis after converting from equations to polynomials. The first polynomial will have seven variables, and we'll choose the six with the largest exponents to be parameters.

eqns = {2 x XX + 2 y YX + 2 z ZX == 2 X,
2 x XY + 2 y YY + 2 z ZY == 2 Y, 2 x XZ + 2 y YZ + 2 z ZZ == 2 Z,
2 X XX + 2 XY Y + 2 XZ Z == 2 x, 2 X YX + 2 Y YY + 2 YZ Z == 2 y,
2 X ZX + 2 Y ZY + 2 Z ZZ == 2 z, 2 x X + 2 YZ ZY - 2 YY ZZ == 2 XX,
2 x Y - 2 YZ ZX + 2 YX ZZ == 2 XY,
2 XZ == 2 x Z + 2 YY ZX - 2 YX ZY,
2 X y - 2 XZ ZY + 2 XY ZZ == 2 YX,
2 y Y + 2 XZ ZX - 2 XX ZZ == 2 YY,
2 YZ == 2 y Z - 2 XY ZX + 2 XX ZY,
2 XZ YY - 2 XY YZ + 2 X z == 2 ZX, -2 XZ YX + 2 XX YZ + 2 Y z ==
2 ZY, 2 XY YX - 2 XX YY + 2 z Z == 2 ZZ,
x^2 + X^2 + XX^2 + XY^2 + XZ^2 + y^2 + Y^2 + YX^2 + YY^2 + YZ^2 +
z^2 + Z^2 + ZX^2 + ZY^2 + ZZ^2 == 3};
polys = Subtract @@@ eqns;
Timing[gb = GroebnerBasis[polys];]
firstvars = Variables[gb[[1]]]
firstexpons = Map[Exponent[gb[[1]], #] &, firstvars]

(* Out[45]= {1.00765, Null}

Out[46]= {XZ, Y, YY, YZ, Z, ZY, ZZ}

Out[47]= {4, 2, 2, 2, 2, 2, 2} *)


Now compute another GB, this time over the field of rational functions in the parameters. We'll see that it contains 9 polynomials.

params = Most[firstvars];
vars = Complement[Variables[polys], params];
Timing[gb2 =
GroebnerBasis[polys, vars, CoefficientDomain -> RationalFunctions];]
gb2 // Length

(* Out[68]= {3.53252, Null}

Out[69]= 9 *)


It's a bit large, with leaf count near 6K. But here goes. But not unpostably large.

In[76]:= gb2

(* Out[76]= {-XZ^4 - XZ^2 Y^2 + XZ^2 YY^2 - XZ^2 YZ^2 + YY^2 YZ^2 -
2 Y YY YZ Z + XZ^2 Z^2 + Y^2 Z^2 +
XZ^2 ZY^2 + (2 YY YZ ZY - 2 Y Z ZY) ZZ + (-XZ^2 +
ZY^2) ZZ^2, -XZ^7 YY^2 - 2 XZ^5 YY^2 YZ^2 - XZ^3 YY^2 YZ^4 +
2 XZ^5 Y YY YZ Z + 2 XZ^3 Y YY YZ^3 Z - XZ^7 Z^2 - XZ^5 Y^2 Z^2 +
XZ^5 YY^2 Z^2 + XZ^7 YY^2 Z^2 + XZ^5 Y^2 YY^2 Z^2 - XZ^5 YY^4 Z^2 -
XZ^5 YZ^2 Z^2 - XZ^3 Y^2 YZ^2 Z^2 + XZ^3 YY^2 YZ^2 Z^2 +
2 XZ^5 YY^2 YZ^2 Z^2 + XZ^3 Y^2 YY^2 YZ^2 Z^2 -
2 XZ^3 YY^4 YZ^2 Z^2 + XZ^3 YY^2 YZ^4 Z^2 - XZ YY^4 YZ^4 Z^2 -
2 XZ^3 Y YY YZ Z^3 - 2 XZ^5 Y YY YZ Z^3 - 2 XZ^3 Y^3 YY YZ Z^3 +
4 XZ^3 Y YY^3 YZ Z^3 - 2 XZ^3 Y YY YZ^3 Z^3 + 4 XZ Y YY^3 YZ^3 Z^3 +
XZ^5 Z^4 + XZ^7 Z^4 + XZ^3 Y^2 Z^4 + 2 XZ^5 Y^2 Z^4 +
XZ^3 Y^4 Z^4 - 2 XZ^5 YY^2 Z^4 - 2 XZ^3 Y^2 YY^2 Z^4 +
XZ^5 YZ^2 Z^4 + XZ^3 Y^2 YZ^2 Z^4 - 2 XZ^3 YY^2 YZ^2 Z^4 -
6 XZ Y^2 YY^2 YZ^2 Z^4 + 4 XZ^3 Y YY YZ Z^5 + 4 XZ Y^3 YY YZ Z^5 -
XZ^5 Z^6 - 2 XZ^3 Y^2 Z^6 - XZ Y^4 Z^6 + XZ^5 Y^2 ZY^2 +
2 XZ^5 YY^2 ZY^2 - XZ^5 Y^2 YY^2 ZY^2 + XZ^5 YZ^2 ZY^2 +
XZ^3 Y^2 YZ^2 ZY^2 - XZ^3 YY^2 YZ^2 ZY^2 - XZ^3 Y^2 YY^2 YZ^2 ZY^2 +
XZ^3 YZ^4 ZY^2 - 3 XZ YY^2 YZ^4 ZY^2 + 4 XZ^3 Y YY YZ Z ZY^2 -
4 XZ^5 Y YY YZ Z ZY^2 - 2 XZ^3 Y^3 YY YZ Z ZY^2 +
4 XZ^3 Y YY^3 YZ Z ZY^2 + 6 XZ Y YY YZ^3 Z ZY^2 -
4 XZ^3 Y YY YZ^3 Z ZY^2 + 4 XZ Y YY^3 YZ^3 Z ZY^2 +
2 XZ^5 Z^2 ZY^2 - 3 XZ^3 Y^2 Z^2 ZY^2 + 2 XZ^5 Y^2 Z^2 ZY^2 +
2 XZ^3 Y^4 Z^2 ZY^2 - XZ^3 YY^2 Z^2 ZY^2 - XZ^5 YY^2 Z^2 ZY^2 -
3 XZ^3 Y^2 YY^2 Z^2 ZY^2 - XZ^5 YZ^2 Z^2 ZY^2 -
3 XZ Y^2 YZ^2 Z^2 ZY^2 + 2 XZ^3 Y^2 YZ^2 Z^2 ZY^2 +
3 XZ YY^2 YZ^2 Z^2 ZY^2 - 11 XZ Y^2 YY^2 YZ^2 Z^2 ZY^2 -
XZ^3 YZ^4 Z^2 ZY^2 + XZ YY^2 YZ^4 Z^2 ZY^2 -
6 XZ Y YY YZ Z^3 ZY^2 + 6 XZ^3 Y YY YZ Z^3 ZY^2 +
10 XZ Y^3 YY YZ Z^3 ZY^2 - 2 XZ Y YY YZ^3 Z^3 ZY^2 - XZ^3 Z^4 ZY^2 -
XZ^5 Z^4 ZY^2 + 3 XZ Y^2 Z^4 ZY^2 - 4 XZ^3 Y^2 Z^4 ZY^2 -
3 XZ Y^4 Z^4 ZY^2 + XZ^3 YZ^2 Z^4 ZY^2 + XZ Y^2 YZ^2 Z^4 ZY^2 -
2 XZ^3 Y^2 ZY^4 + XZ^3 Y^4 ZY^4 - XZ^3 YY^2 ZY^4 +
XZ^3 Y^2 YY^2 ZY^4 - 2 XZ^3 YZ^2 ZY^4 - XZ Y^2 YZ^2 ZY^4 +
XZ^3 Y^2 YZ^2 ZY^4 + 3 XZ YY^2 YZ^2 ZY^4 - 3 XZ Y^2 YY^2 YZ^2 ZY^4 -
XZ YZ^4 ZY^4 - 6 XZ Y YY YZ Z ZY^4 + 4 XZ^3 Y YY YZ Z ZY^4 +
6 XZ Y^3 YY YZ Z ZY^4 - XZ^3 Z^2 ZY^4 + 4 XZ Y^2 Z^2 ZY^4 -
2 XZ^3 Y^2 Z^2 ZY^4 - 3 XZ Y^4 Z^2 ZY^4 + XZ YZ^2 Z^2 ZY^4 +
XZ^3 YZ^2 Z^2 ZY^4 + XZ Y^2 ZY^6 - XZ Y^4 ZY^6 + XZ YZ^2 ZY^6 -
XZ Y^2 YZ^2 ZY^6 +
ZX^2 (-XZ^5 YY^4 - 2 XZ^3 YY^4 YZ^2 - XZ YY^4 YZ^4 +
4 XZ^3 Y YY^3 YZ Z + 4 XZ Y YY^3 YZ^3 Z - 2 XZ^5 YY^2 Z^2 -
2 XZ^3 Y^2 YY^2 Z^2 - 2 XZ^3 YY^2 YZ^2 Z^2 -
6 XZ Y^2 YY^2 YZ^2 Z^2 + 4 XZ^3 Y YY YZ Z^3 +
4 XZ Y^3 YY YZ Z^3 - XZ^5 Z^4 - 2 XZ^3 Y^2 Z^4 - XZ Y^4 Z^4 +
2 XZ^3 Y^2 YY^2 ZY^2 - 2 XZ^3 YY^2 YZ^2 ZY^2 -
2 XZ Y^2 YY^2 YZ^2 ZY^2 - 2 XZ YY^2 YZ^4 ZY^2 +
8 XZ^3 Y YY YZ Z ZY^2 + 4 XZ Y^3 YY YZ Z ZY^2 +
4 XZ Y YY YZ^3 Z ZY^2 - 2 XZ^3 Y^2 Z^2 ZY^2 -
2 XZ Y^4 Z^2 ZY^2 + 2 XZ^3 YZ^2 Z^2 ZY^2 -
2 XZ Y^2 YZ^2 Z^2 ZY^2 - XZ Y^4 ZY^4 - 2 XZ Y^2 YZ^2 ZY^4 -
XZ YZ^4 ZY^4) +
ZX (2 XZ^6 YY^3 + 4 XZ^4 YY^3 YZ^2 + 2 XZ^2 YY^3 YZ^4 -
6 XZ^4 Y YY^2 YZ Z - 6 XZ^2 Y YY^2 YZ^3 Z + 2 XZ^6 YY Z^2 +
2 XZ^4 Y^2 YY Z^2 + 2 XZ^4 YY YZ^2 Z^2 +
6 XZ^2 Y^2 YY YZ^2 Z^2 - 2 XZ^4 Y YZ Z^3 - 2 XZ^2 Y^3 YZ Z^3 -
2 XZ^4 Y^2 YY ZY^2 - 2 XZ^4 YY^3 ZY^2 + 2 XZ^4 YY YZ^2 ZY^2 +
2 XZ^2 Y^2 YY YZ^2 ZY^2 + 2 XZ^2 YY YZ^4 ZY^2 +
2 YY^3 YZ^4 ZY^2 - 4 XZ^4 Y YZ Z ZY^2 - 2 XZ^2 Y^3 YZ Z ZY^2 -
6 XZ^2 Y YY^2 YZ Z ZY^2 - 2 XZ^2 Y YZ^3 Z ZY^2 -
6 Y YY^2 YZ^3 Z ZY^2 - 2 XZ^4 YY Z^2 ZY^2 +
6 XZ^2 Y^2 YY Z^2 ZY^2 - 6 XZ^2 YY YZ^2 Z^2 ZY^2 +
6 Y^2 YY YZ^2 Z^2 ZY^2 + 6 XZ^2 Y YZ Z^3 ZY^2 -
2 Y^3 YZ Z^3 ZY^2 + 2 XZ^2 Y^2 YY ZY^4 - 2 XZ^2 YY YZ^2 ZY^4 +
2 Y^2 YY YZ^2 ZY^4 + 2 YY YZ^4 ZY^4 + 4 XZ^2 Y YZ Z ZY^4 -
2 Y^3 YZ Z ZY^4 - 2 Y YZ^3 Z ZY^4) +
ZX (-2 XZ^4 YY^2 YZ ZY - 2 XZ^2 YY^2 YZ^3 ZY + 4 XZ^4 Y YY Z ZY +
4 XZ^2 Y YY YZ^2 Z ZY + 2 XZ^4 YZ Z^2 ZY -
2 XZ^2 Y^2 YZ Z^2 ZY - 2 XZ^2 Y^2 YZ ZY^3 +
2 XZ^2 YY^2 YZ ZY^3 - 2 XZ^2 YZ^3 ZY^3 + 2 YY^2 YZ^3 ZY^3 -
4 XZ^2 Y YY Z ZY^3 - 4 Y YY YZ^2 Z ZY^3 - 2 XZ^2 YZ Z^2 ZY^3 +
2 Y^2 YZ Z^2 ZY^3 + 2 Y^2 YZ ZY^5 +
2 YZ^3 ZY^5) ZZ + (2 XZ^5 YY YZ ZY + 2 XZ^3 YY YZ^3 ZY -
2 XZ^5 Y Z ZY - 2 XZ^5 Y YY^2 Z ZY - 2 XZ^3 Y YZ^2 Z ZY -
2 XZ^3 Y YY^2 YZ^2 Z ZY - 2 XZ^3 YY YZ Z^2 ZY -
2 XZ^5 YY YZ Z^2 ZY + 2 XZ^3 Y^2 YY YZ Z^2 ZY -
2 XZ^3 YY YZ^3 Z^2 ZY + 2 XZ^3 Y Z^3 ZY + 2 XZ^3 Y YZ^2 Z^3 ZY -
4 XZ^3 YY YZ ZY^3 + 2 XZ^3 Y^2 YY YZ ZY^3 - 2 XZ YY YZ^3 ZY^3 +
4 XZ^3 Y Z ZY^3 + 2 XZ^3 Y YY^2 Z ZY^3 + 2 XZ Y YZ^2 Z ZY^3 +
2 XZ^3 Y YZ^2 Z ZY^3 + 2 XZ Y YY^2 YZ^2 Z ZY^3 +
2 XZ YY YZ Z^2 ZY^3 + 2 XZ^3 YY YZ Z^2 ZY^3 -
2 XZ Y^2 YY YZ Z^2 ZY^3 + 2 XZ YY YZ^3 Z^2 ZY^3 -
2 XZ Y Z^3 ZY^3 - 2 XZ Y YZ^2 Z^3 ZY^3 + 2 XZ YY YZ ZY^5 -
2 XZ Y^2 YY YZ ZY^5 - 2 XZ Y Z ZY^5 -
2 XZ Y YZ^2 Z ZY^5) ZZ, -XZ^5 Y ZY - XZ^3 Y YZ^2 ZY -
2 XZ^3 YY YZ Z ZY - 2 XZ YY YZ^3 Z ZY + 3 XZ^3 Y Z^2 ZY +
2 XZ Y YZ^2 Z^2 ZY + 2 XZ YY YZ Z^3 ZY - 2 XZ Y Z^4 ZY +
2 XZ^3 Y ZY^3 + XZ Y YZ^2 ZY^3 + 2 XZ YY YZ Z ZY^3 -
3 XZ Y Z^2 ZY^3 - XZ Y ZY^5 +
ZX (XZ^4 Y YY ZY + XZ^2 Y YY YZ^2 ZY + XZ^4 YZ Z ZY +
XZ^2 YY^2 YZ Z ZY + XZ^2 YZ^3 Z ZY + YY^2 YZ^3 Z ZY -
2 XZ^2 Y YY Z^2 ZY - 2 Y YY YZ^2 Z^2 ZY - XZ^2 YZ Z^3 ZY +
Y^2 YZ Z^3 ZY - XZ^2 Y YY ZY^3 - Y YY YZ^2 ZY^3 -
XZ^2 YZ Z ZY^3 + Y^2 YZ Z ZY^3) +
z (-XZ^5 Z^2 - XZ^3 Y^2 Z^2 + XZ^3 YY^2 Z^2 - XZ^3 YZ^2 Z^2 +
XZ YY^2 YZ^2 Z^2 - 2 XZ Y YY YZ Z^3 + XZ^3 Z^4 + XZ Y^2 Z^4 -
XZ^3 Y^2 ZY^2 - 2 XZ Y YY YZ Z ZY^2 + XZ^3 Z^2 ZY^2 +
2 XZ Y^2 Z^2 ZY^2 + XZ Y^2 ZY^4) +
ZX (-XZ^4 YY Z - XZ^2 YY YZ^2 Z + XZ^2 Y YZ Z^2 + XZ^2 Y YZ ZY^2 +
XZ^2 YY Z ZY^2 + YY YZ^2 Z ZY^2 - Y YZ Z^2 ZY^2 -
Y YZ ZY^4) ZZ + (XZ^5 Z + XZ^3 YZ^2 Z - XZ^3 Z^3 -
2 XZ^3 Z ZY^2 - XZ YZ^2 Z ZY^2 + XZ Z^3 ZY^2 + XZ Z ZY^4) ZZ,
XZ^4 Y^2 ZY + 3 XZ^2 Y YY YZ Z ZY + XZ^4 Z^2 ZY - 2 XZ^2 Y^2 Z^2 ZY -
XZ^2 YY^2 Z^2 ZY + XZ^2 YZ^2 Z^2 ZY + YY^2 YZ^2 Z^2 ZY -
2 Y YY YZ Z^3 ZY - XZ^2 Z^4 ZY + Y^2 Z^4 ZY - XZ^2 Y^2 ZY^3 -
Y YY YZ Z ZY^3 - XZ^2 Z^2 ZY^3 + Y^2 Z^2 ZY^3 +
ZX (-XZ^3 Y^2 YY ZY - XZ^3 Y YZ Z ZY - 2 XZ Y YY^2 YZ Z ZY +
2 XZ Y^2 YY Z^2 ZY - 2 XZ YY YZ^2 Z^2 ZY + 2 XZ Y YZ Z^3 ZY +
XZ Y^2 YY ZY^3 + XZ Y YZ Z ZY^3) +
YX (XZ^5 Z^2 + XZ^3 Y^2 Z^2 - XZ^3 YY^2 Z^2 + XZ^3 YZ^2 Z^2 -
XZ YY^2 YZ^2 Z^2 + 2 XZ Y YY YZ Z^3 - XZ^3 Z^4 - XZ Y^2 Z^4 +
XZ^3 Y^2 ZY^2 + 2 XZ Y YY YZ Z ZY^2 - XZ^3 Z^2 ZY^2 -
2 XZ Y^2 Z^2 ZY^2 - XZ Y^2 ZY^4) +
ZX (XZ^3 Y YY Z + XZ^3 YZ Z^2 - XZ Y YY Z ZY^2 -
XZ YZ Z^2 ZY^2) ZZ + (-XZ^4 Y Z - XZ^2 YY YZ Z^2 + XZ^2 Y Z^3 +
2 XZ^2 Y Z ZY^2 + YY YZ Z^2 ZY^2 - Y Z^3 ZY^2 - Y Z ZY^4) ZZ,
XZ^5 YZ Z + XZ^3 Y^2 YZ Z - XZ^3 YY^2 YZ Z + XZ^3 YZ^3 Z -
XZ YY^2 YZ^3 Z + 2 XZ Y YY YZ^2 Z^2 - XZ^3 YZ Z^3 - XZ Y^2 YZ Z^3 +
XZ^3 Y YY ZY^2 - XZ^3 YZ Z ZY^2 + 2 XZ YY^2 YZ Z ZY^2 -
2 XZ Y YY Z^2 ZY^2 - XZ Y YY ZY^4 +
ZX (-XZ^4 YY YZ Z - XZ^2 Y^2 YY YZ Z + XZ^2 YY^3 YZ Z -
XZ^2 YY YZ^3 Z + YY^3 YZ^3 Z + XZ^4 Y Z^2 + XZ^2 Y^3 Z^2 -
XZ^2 Y YY^2 Z^2 + XZ^2 Y YZ^2 Z^2 - 3 Y YY^2 YZ^2 Z^2 +
XZ^2 YY YZ Z^3 + 3 Y^2 YY YZ Z^3 - XZ^2 Y Z^4 - Y^3 Z^4 +
XZ^2 Y^3 ZY^2 - XZ^2 Y YY^2 ZY^2 + XZ^2 Y YZ^2 ZY^2 -
Y YY^2 YZ^2 ZY^2 - XZ^2 YY YZ Z ZY^2 + 2 Y^2 YY YZ Z ZY^2 -
Y^3 Z^2 ZY^2) +
y (-XZ^5 Z^2 - XZ^3 Y^2 Z^2 + XZ^3 YY^2 Z^2 - XZ^3 YZ^2 Z^2 +
XZ YY^2 YZ^2 Z^2 - 2 XZ Y YY YZ Z^3 + XZ^3 Z^4 + XZ Y^2 Z^4 -
XZ^3 Y^2 ZY^2 - 2 XZ Y YY YZ Z ZY^2 + XZ^3 Z^2 ZY^2 +
2 XZ Y^2 Z^2 ZY^2 + XZ Y^2 ZY^4) +
ZX (-XZ^2 Y YY YZ ZY + XZ^2 YY^2 Z ZY - XZ^2 YZ^2 Z ZY +
YY^2 YZ^2 Z ZY - 2 Y YY YZ Z^2 ZY + XZ^2 Z^3 ZY + Y^2 Z^3 ZY -
Y YY YZ ZY^3 + Y^2 Z ZY^3) ZZ + (XZ^3 Y YZ ZY - XZ^3 YY Z ZY -
XZ Y YZ ZY^3 + XZ YY Z ZY^3) ZZ, -XY XZ - YY YZ + Y Z -
ZY ZZ, -XZ^3 Y^2 YZ ZY + XZ^3 Y YY Z ZY - 2 XZ Y YY YZ^2 Z ZY +
2 XZ Y^2 YZ Z^2 ZY + 2 XZ YY^2 YZ Z^2 ZY - 2 XZ Y YY Z^3 ZY +
XZ Y^2 YZ ZY^3 - XZ Y YY Z ZY^3 +
ZX (XZ^2 Y^2 YY YZ ZY - XZ^2 Y YY^2 Z ZY + XZ^2 Y YZ^2 Z ZY +
Y YY^2 YZ^2 Z ZY - 2 XZ^2 YY YZ Z^2 ZY - 2 Y^2 YY YZ Z^2 ZY +
XZ^2 Y Z^3 ZY + Y^3 Z^3 ZY - Y^2 YY YZ ZY^3 + Y^3 Z ZY^3) +
XX (XZ^5 Z^2 + XZ^3 Y^2 Z^2 - XZ^3 YY^2 Z^2 + XZ^3 YZ^2 Z^2 -
XZ YY^2 YZ^2 Z^2 + 2 XZ Y YY YZ Z^3 - XZ^3 Z^4 - XZ Y^2 Z^4 +
XZ^3 Y^2 ZY^2 + 2 XZ Y YY YZ Z ZY^2 - XZ^3 Z^2 ZY^2 -
2 XZ Y^2 Z^2 ZY^2 - XZ Y^2 ZY^4) + (XZ^3 Y YZ Z - XZ^3 YY Z^2 -
XZ Y YZ Z ZY^2 + XZ YY Z^2 ZY^2) ZZ +
ZX (-XZ^2 Y YY YZ Z + XZ^4 Z^2 + XZ^2 Y^2 Z^2 + XZ^2 Y^2 ZY^2 +
Y YY YZ Z ZY^2 - XZ^2 Z^2 ZY^2 - Y^2 Z^2 ZY^2 -
Y^2 ZY^4) ZZ, -XZ^4 Y YY ZY - XZ^2 Y YY YZ^2 ZY - XZ^4 YZ Z ZY -
XZ^2 YY^2 YZ Z ZY - XZ^2 YZ^3 Z ZY - YY^2 YZ^3 Z ZY +
2 XZ^2 Y YY Z^2 ZY + 2 Y YY YZ^2 Z^2 ZY + XZ^2 YZ Z^3 ZY -
Y^2 YZ Z^3 ZY + XZ^2 Y YY ZY^3 + Y YY YZ^2 ZY^3 + XZ^2 YZ Z ZY^3 -
Y^2 YZ Z ZY^3 +
ZX (XZ^3 Y YY^2 ZY + XZ Y YY^2 YZ^2 ZY + 2 XZ^3 YY YZ Z ZY +
2 XZ YY YZ^3 Z ZY - XZ^3 Y Z^2 ZY - XZ Y^3 Z^2 ZY -
2 XZ Y YZ^2 Z^2 ZY - XZ Y^3 ZY^3 - XZ Y YZ^2 ZY^3) +
X (-XZ^5 Z^2 - XZ^3 Y^2 Z^2 + XZ^3 YY^2 Z^2 - XZ^3 YZ^2 Z^2 +
XZ YY^2 YZ^2 Z^2 - 2 XZ Y YY YZ Z^3 + XZ^3 Z^4 + XZ Y^2 Z^4 -
XZ^3 Y^2 ZY^2 - 2 XZ Y YY YZ Z ZY^2 + XZ^3 Z^2 ZY^2 +
2 XZ Y^2 Z^2 ZY^2 + XZ Y^2 ZY^4) +
ZX (-XZ^3 YY^2 Z - XZ YY^2 YZ^2 Z + 2 XZ Y YY YZ Z^2 - XZ^3 Z^3 -
XZ Y^2 Z^3 + 2 XZ Y YY YZ ZY^2 - XZ Y^2 Z ZY^2 +
XZ YZ^2 Z ZY^2) ZZ + (XZ^4 YY Z + XZ^2 YY YZ^2 Z -
XZ^2 Y YZ Z^2 - XZ^2 Y YZ ZY^2 - XZ^2 YY Z ZY^2 -
YY YZ^2 Z ZY^2 + Y YZ Z^2 ZY^2 + Y YZ ZY^4) ZZ,
XZ^6 Z + XZ^4 Y^2 Z - XZ^4 YY^2 Z + XZ^4 YZ^2 Z - XZ^2 YY^2 YZ^2 Z +
2 XZ^2 Y YY YZ Z^2 - XZ^4 Z^3 - XZ^2 Y^2 Z^3 - XZ^2 Y YY YZ ZY^2 -
2 XZ^4 Z ZY^2 + XZ^2 YY^2 Z ZY^2 - XZ^2 YZ^2 Z ZY^2 -
YY^2 YZ^2 Z ZY^2 + 2 Y YY YZ Z^2 ZY^2 + XZ^2 Z^3 ZY^2 -
Y^2 Z^3 ZY^2 + Y YY YZ ZY^4 + XZ^2 Z ZY^4 - Y^2 Z ZY^4 +
x (-XZ^5 Z^2 - XZ^3 Y^2 Z^2 + XZ^3 YY^2 Z^2 - XZ^3 YZ^2 Z^2 +
XZ YY^2 YZ^2 Z^2 - 2 XZ Y YY YZ Z^3 + XZ^3 Z^4 + XZ Y^2 Z^4 -
XZ^3 Y^2 ZY^2 - 2 XZ Y YY YZ Z ZY^2 + XZ^3 Z^2 ZY^2 +
2 XZ Y^2 Z^2 ZY^2 + XZ Y^2 ZY^4) +
ZX (-XZ^5 YY Z - XZ^3 Y^2 YY Z + XZ^3 YY^3 Z - XZ^3 YY YZ^2 Z +
XZ YY^3 YZ^2 Z - 2 XZ Y YY^2 YZ Z^2 + XZ^3 YY Z^3 +
XZ Y^2 YY Z^3 + XZ^3 Y YZ ZY^2 + XZ^3 YY Z ZY^2 +
2 XZ YY YZ^2 Z ZY^2 - 2 XZ Y YZ Z^2 ZY^2 - XZ Y YZ ZY^4) +
ZX (-XZ^3 Y YY ZY - XZ^3 YZ Z ZY + XZ Y YY ZY^3 +
XZ YZ Z ZY^3) ZZ + (XZ^4 Y ZY + XZ^2 YY YZ Z ZY -
XZ^2 Y Z^2 ZY - 2 XZ^2 Y ZY^3 - YY YZ Z ZY^3 + Y Z^2 ZY^3 +
Y ZY^5) ZZ} *)

• +1 for this approach. Dec 16, 2021 at 19:44
• Your code contains GroebnerBasis[polys], which is inconsistent with the documentation and, according to the Mathematica frontend, is missing an argument. Nonetheless, it works., although it produces a different result from GroebnerBasis[polys, Variables[polys]];. Can you provide an explanation, or would you prefer that I post a question, so that your explanation has more visibility? Thanks. Dec 17, 2021 at 3:18
• @bbgodfrey This is an uncommon case where I don't know what variable order to use, so I let GB choose. I probably should have added Sort->True (and specified the variables in the way you suggest). Though that can backfire, since the heuristics behind this sorting are fallible. Dec 17, 2021 at 14:26

The following provides a solution to the sixteen equations for nine dependent variables in terms of six independent variables, and from this the desired nine equations can be obtained. With the sixteen equations designated eqs,

s = Solve[eqs, {YZ, x, X, YY, Y, z, XY, XZ, y}]


produces a result in several minutes. With

s // Dimensions
(* {4, 9} *)
s // LeafCount
(* 1550453 *)


it is far too large to reproduce here. The nine corresponding equations follow immediately from any of the four solution sets, for instance.

First[s] /. Rule -> Equal


Whether this ponderous result is of any practical value is unclear to me.

The obvious question to ask is, why choose to solve for {YZ, x, X, YY, Y, z, XY, XZ, y} from among some 5000 other possibilities. In fact, it was by trial and error. I found a readily solvable set after only a few tries, suggesting that the set of six independent variables is not at all unique.