Fast expansion of a polynomial with R

Posted on July 4, 2023 by Stéphane Laurent

Tags: R, maths

In this previous post, this previous post, and this previous post, I showed how to expand a polynomial with symbolic parameters. In the first two posts, I used the R package caracas, a wrapper of SymPy, and in the third post I used Julia.

Now I’ve found a pure R solution, with the help of the spray package, and it is very fast. That’s what I’m going to demonstrate here, with the same example I treated with Julia.

We will lost something as compared to Julia: the spray package does not handle rational numbers, so we will replace them by their double approximation. We could use the qspray package instead, to deal with the rational numbers.

Here is the starting polynomial expression:

# define the polynomial expression as a function
f <- function(x, y, z, w, sqrt3) {
  ((x^2+y^2+z^2+w^2+145/3)^2-4*(9*z^2+16*w^2))^2 * 
    ((x^2+y^2+z^2+w^2+145/3)^2+296*(x^2+y^2)-4*(9*z^2+16*w^2)) - 
    16*(x^2+y^2)*(x^2+y^2+z^2+w^2+145/3)^2 * 
    (37*(x^2+y^2+z^2+w^2+145/3)^2-1369*(x^2+y^2)-7*(225*z^2+448*w^2)) - 
    sqrt3*16/9*(x^3-3*x*y^2) * 
    (110*(x^2+y^2+z^2+w^2+145/3)^3-148*(x^2+y^2+z^2+w^2+145/3) * 
       (110*x^2+110*y^2-297*z^2+480*w^2)) - 
    64*(x^2+y^2)*(3*(729*z^4+4096*w^4)+168*(x^2+y^2)*(15*z^2-22*w^2)) + 
    64*(12100/27*(x^3-3*x*y^2)^2-7056*(3*x^2*y-y^3)^2)-592240896*z^2*w^2
}

We transform it to the polynomial of interest:

# there are 3 variables (x,y,z) and 6 parameters (w0,a,b,c,d,sqrt3)
library(spray)
x     <- lone(1, 9)
y     <- lone(2, 9)
z     <- lone(3, 9)
w0    <- lone(4, 9)
a     <- lone(5, 9)
b     <- lone(6, 9)
c     <- lone(7, 9)
d     <- lone(8, 9)
sqrt3 <- lone(9, 9)

# define the substitutions
X <- a*x  - b*y  - c*z  - d*w0
Y <- a*y  + b*x  + c*w0 - d*z
Z <- a*z  - b*w0 + c*x  + d*y
W <- a*w0 + b*z  - c*y  + d*x

# here is the polynomial of interest
P <- f(X, Y, Z, W, sqrt3)

Now we take the “xyz” part of each term:

# the exponents of x, y, z
xyz_powers <- P[["index"]][, c(1L, 2L, 3L)]

# the "keys" xyz(i,j,k) for POV-Ray
xyz_povray <- apply(xyz_powers, 1L, function(comp) {
  sprintf("xyz(%s): ", toString(comp))
})

Now we take the remaining part of each term:

# the other exponents, those of w0, a, b, c, d, sqrt3
other_powers <- P[["index"]][, c(4L, 5L, 6L, 7L, 8L, 9L)]

# the polynomials in w0, a, b, c, d, sqrt3
nterms <- length(P)
coeffs <- P[["value"]]
polynomials <- lapply(1L:nterms, function(i) {
  spray(other_powers[i, ], coeffs[i])
})

And we group these polynomials, putting together those which share the same “xyz” part:

# group the polynomials which have the same x,y,z exponents
polynomials_groups <- sapply(split(polynomials, xyz_povray), function(polys) {
  polysum <- polys[[1]]
  for(poly in polys[-1]) {
    polysum <- spray_add(
      polysum$index, polysum$value, poly$index, poly$value
    )
  }
  as.spray(polysum)
}, simplify = FALSE)
# remove the empty polynomials, if there are some
polynomials_groups <- Filter(Negate(is.empty), polynomials_groups)

We want these polynomials as character strings:

# there's no as.character function for sprays (polynomials), so we make one
asCharacter <- function(poly) {
  op <- options(sprayvars = c("w0", "a", "b", "c", "d", "sqrt3"))
  x <- capture.output(print_spray_polyform(poly))
  options(op)
  paste0(x, collapse = " ")
}

polynomials_strings <- 
  sapply(polynomials_groups, asCharacter, simplify = FALSE)

And that’s it. We have everything needed to construct the POV-Ray code:

head(polynomials_strings, 2L)
## $`xyz(0, 0, 0): `
## [1] "12749128107.8532 +1760*w0^7*c^6*d*sqrt3 -6*w0^10*d^10 +w0^12*b^12 +60*w0^12*b^2*c^4*d^6 +180*w0^12*a^4*b^2*c^2*d^4 +9498*w0^8*b^4*c^4 +44148*w0^8*a^4*b^2*d^2 +180*w0^12*a^4*b^2*c^4*d^2 +1068*w0^10*a^4*c^4*d^2 +30*w0^12*a^2*c^8*d^2 +2196*w0^10*a^4*b^2*d^4 +2519250.2222222*w0^6*a^2*b^4 +10209.6666667*w0^8*a^8 +6*w0^12*a^10*d^2 -12537.3333333*w0^8*a^2*d^6 +31964*w0^8*b^6*d^2 +2076*w0^10*b^4*c^4*d^2 +182*w0^10*b^10 +296*w0^10*a^2*c^6*d^2 +w0^12*a^12 +593516.8888889*w0^6*b^4*d^2 +2076*w0^10*b^4*c^2*d^4 -2933.3333333*w0^9*a^2*c^4*d^3*sqrt3 -1345147.5555556*w0^6*a^2*c^2*d^2 -1760*w0^9*a^2*c^6*d*sqrt3 -4697.3333333*w0^8*b^2*c^6 +18996*w0^8*b^4*c^2*d^2 -14022*w0^8*a^4*c^4 +386*w0^10*a^8*d^2 +6*w0^12*a^10*c^2 -317738.6666667*w0^7*b^2*c^4*d*sqrt3 -60*w0^10*c^6*d^4 +15*w0^12*a^4*c^8 +2136*w0^10*b^6*c^2*d^2 +180*w0^12*a^2*b^4*c^4*d^2 +60*w0^12*a^6*b^2*c^4 -6876*w0^8*a^2*b^2*c^4 +67668*w0^8*a^2*b^4*d^2 +60*w0^12*b^4*c^6*d^2 +15*w0^12*a^8*b^4 +30*w0^12*a^8*c^2*d^2 +20*w0^12*a^6*c^6 +632*w0^10*b^2*c^2*d^6 +722*w0^10*b^8*c^2 +67668*w0^8*a^2*b^4*c^2 +120*w0^12*a^6*b^2*c^2*d^2 -28355.5555556*w0^5*d^5*sqrt3 -37612*w0^8*a^2*c^2*d^4 +6*w0^12*a^2*d^10 -1124*w0^8*c^6*d^2 +1105545.7777778*w0^6*a^2*b^2*c^2 +722*w0^10*b^8*d^2 -586.6666667*w0^9*a^6*c^2*d*sqrt3 +6*w0^12*a^10*b^2 +2933.3333333*w0^7*c^4*d^3*sqrt3 -1173.3333333*w0^9*c^4*d^5*sqrt3 -1760*w0^9*a^2*b^4*c^2*d*sqrt3 +6*w0^12*a^2*b^10 +60*w0^12*a^2*b^4*d^6 +3144*w0^10*a^2*b^2*c^2*d^4 +30*w0^12*a^8*b^2*d^2 +30*w0^12*a^2*b^2*d^8 +356*w0^10*a^4*c^6 +1484*w0^10*a^4*b^6 +56711.1111111*w0^5*c^2*d^3*sqrt3 +44148*w0^8*a^4*b^2*c^2 +90*w0^12*a^4*c^4*d^4 +6*w0^12*a^2*c^10 -1124*w0^8*c^2*d^6 -37612*w0^8*a^2*c^4*d^2 +564*w0^10*a^6*c^4 -4733659.2592593*w0^5*a^2*d^3*sqrt3 +2552*w0^10*a^2*b^6*c^2 +15*w0^12*b^8*c^4 +85066.6666667*w0^5*c^4*d*sqrt3 +60*w0^12*a^2*b^6*c^4 +60*w0^12*b^6*c^2*d^4 -3520*w0^9*a^2*b^2*c^4*d*sqrt3 +1173.3333333*w0^9*a^2*b^2*d^5*sqrt3 +15*w0^12*a^4*d^8 +30*w0^12*a^8*b^2*c^2 +18049.6666667*w0^8*b^8 -1409995.5555556*w0^6*b^2*c^2*d^2 +w0^12*c^12 +60*w0^12*b^6*c^4*d^2 -14022*w0^8*a^4*d^4 +w0^12*d^12 +98*w0^10*a^10 +1048*w0^10*a^2*b^2*c^6 +90*w0^12*a^4*b^4*d^4 +60*w0^12*a^6*b^2*d^4 +5147478.5185185*w0^5*b^2*d^3*sqrt3 +31964*w0^8*b^6*c^2 +993249552.469136*w0^2*b^2 -281*w0^8*d^8 +30*w0^12*a^2*b^2*c^8 +90*w0^12*a^4*b^4*c^4 -656447.2222222*w0^4*d^4 +8444*w0^8*a^6*c^2 -211825.7777778*w0^7*b^2*c^2*d^3*sqrt3 +586.6666667*w0^7*c^2*d^5*sqrt3 +15*w0^12*b^4*d^8 +1073624.7407407*w0^6*b^6 -1760*w0^9*a^4*b^2*c^2*d*sqrt3 -66241728.3950617*w0^3*c^2*d*sqrt3 -1760*w0^9*b^2*c^6*d*sqrt3 +20*w0^12*a^6*b^6 -586.6666667*w0^9*b^6*c^2*d*sqrt3 +60*w0^12*b^2*c^6*d^4 +106499.5555556*w0^7*b^4*d^3*sqrt3 -656447.2222222*w0^4*c^4 -586.6666667*w0^9*b^2*c^2*d^5*sqrt3 +30*w0^12*a^2*b^8*c^2 +60*w0^12*a^4*b^6*d^2 -9660598.1481482*w0^4*b^2*d^2 -12537.3333333*w0^8*a^2*c^6 -97937.7777778*w0^7*a^4*d^3*sqrt3 -672573.7777778*w0^6*a^2*c^4 +586.6666667*w0^9*a^4*b^2*d^3*sqrt3 -1312894.4444444*w0^4*c^2*d^2 +3324*w0^10*a^4*b^4*d^2 +60*w0^12*a^4*b^6*c^2 +1880*w0^10*a^6*b^2*d^2 +586.6666667*w0^9*a^2*b^4*d^3*sqrt3 +5400*w0^10*a^2*b^4*c^2*d^2 +1048*w0^10*a^2*b^2*d^6 +23850915.7407407*w0^4*a^4 +60*w0^12*a^2*b^6*d^4 +60*w0^12*a^4*b^2*c^6 +586.6666667*w0^9*a^4*d^5*sqrt3 -27975709.2592593*w0^4*a^2*d^2 -30*w0^10*c^2*d^8 +60*w0^12*a^2*c^6*d^4 -30*w0^10*c^8*d^2 -704997.7777778*w0^6*b^2*d^4 +30*w0^12*a^2*b^8*d^2 +120*w0^12*a^2*b^2*c^6*d^2 +46326.6666667*w0^8*a^6*b^2 +255375.5555556*w0^6*d^6 +1068*w0^10*b^6*d^4 +20*w0^12*c^6*d^6 +60*w0^12*a^6*b^4*d^2 -224889.9259259*w0^6*c^6 +120*w0^12*a^2*b^2*c^2*d^6 +60*w0^12*a^6*b^4*c^2 +62006.6666667*w0^8*a^2*b^6 +8561.7777778*w0^7*a^2*b^2*d^3*sqrt3 +492736.7407407*w0^6*a^6 +60*w0^12*a^2*c^4*d^6 +3647719.5555556*w0^6*c^4*d^2 +60*w0^12*a^4*c^2*d^6 +20*w0^12*b^6*c^6 +15*w0^12*c^4*d^8 +15*w0^12*b^8*d^4 +6*w0^12*c^2*d^10 -14092*w0^8*b^2*c^2*d^4 +586.6666667*w0^9*b^2*d^7*sqrt3 +158*w0^10*b^2*d^8 +2700*w0^10*a^2*b^4*d^4 -32744490.7407408*w0^2*c^2 +586.6666667*w0^9*a^2*d^7*sqrt3 +692*w0^10*b^4*c^6 +90*w0^12*b^4*c^4*d^4 -704997.7777778*w0^6*b^2*c^4 +593516.8888889*w0^6*b^4*c^2 +195.5555556*w0^9*d^9*sqrt3 -25685.3333333*w0^7*a^2*b^2*c^2*d*sqrt3 +45052.8888889*w0^6*a^4*c^2 -6*w0^10*c^10 -319498.6666667*w0^7*b^4*c^2*d*sqrt3 +42166026.8518519*w0^4*b^4 +195.5555556*w0^9*b^6*d^3*sqrt3 +1105545.7777778*w0^6*a^2*b^2*d^2 +293813.3333333*w0^7*a^4*c^2*d*sqrt3 -586.6666667*w0^9*c^8*d*sqrt3 -531718486.740741*w0^4*a^2*b^2 +120*w0^12*a^2*b^6*c^2*d^2 -2933.3333333*w0^9*b^2*c^4*d^3*sqrt3 +2552*w0^10*a^2*b^6*d^2 +296*w0^10*a^2*c^2*d^6 +2700*w0^10*a^2*b^4*c^4 -32744490.7407408*w0^2*d^2 +6*w0^12*b^2*d^10 +444*w0^10*a^2*c^4*d^4 +3144*w0^10*a^2*b^2*c^4*d^2 +105912.8888889*w0^7*b^2*d^5*sqrt3 +195.5555556*w0^9*a^6*d^3*sqrt3 -3556262.6666667*w0^6*c^2*d^4 +180*w0^12*a^4*b^4*c^2*d^2 -4697.3333333*w0^8*b^2*d^6 +2196*w0^10*a^4*b^2*c^4 -15442435.5555556*w0^5*b^2*c^2*d*sqrt3 +6*w0^12*b^2*c^10 -60*w0^10*c^4*d^6 +158*w0^10*b^2*c^8 -1564.4444444*w0^9*c^6*d^3*sqrt3 +1880*w0^10*a^6*b^2*c^2 +295573.3333333*w0^7*a^2*c^4*d*sqrt3 +632*w0^10*b^2*c^6*d^2 +6*w0^12*b^10*d^2 +6*w0^12*b^10*c^2 +1068*w0^10*b^6*c^4 +60*w0^12*b^4*c^2*d^6 -1173.3333333*w0^9*a^4*c^2*d^3*sqrt3 +14200977.7777778*w0^5*a^2*c^2*d*sqrt3 +8444*w0^8*a^6*d^2 +564*w0^10*a^6*d^4 +15*w0^12*c^8*d^4 +9498*w0^8*b^4*d^4 +15*w0^12*a^8*d^4 +574*w0^10*a^8*b^2 -27975709.2592593*w0^4*a^2*c^2 +15*w0^12*b^4*c^8 +15*w0^12*a^4*b^8 -9660598.1481482*w0^4*b^2*c^2 +534826682.098766*w0^2*a^2 +60*w0^12*a^2*b^4*c^6 +180*w0^12*a^2*b^2*c^4*d^4 -98524.4444444*w0^7*a^2*d^5*sqrt3 +60*w0^12*a^6*c^2*d^4 +20*w0^12*b^6*d^6 +1960314.2222222*w0^6*a^4*b^2 +586.6666667*w0^9*b^4*d^5*sqrt3 +1068*w0^10*a^4*c^2*d^4 -1173.3333333*w0^9*b^4*c^2*d^3*sqrt3 +80074*w0^8*a^4*b^4 +20*w0^12*a^6*d^6 +74*w0^10*a^2*c^8 -14092*w0^8*b^2*c^4*d^2 +948*w0^10*b^2*c^4*d^4 +3324*w0^10*a^4*b^4*c^2 +30*w0^12*b^2*c^8*d^2 -586.6666667*w0^7*d^7*sqrt3 -1760*w0^9*b^4*c^4*d*sqrt3 +30*w0^12*b^2*c^2*d^8 +30*w0^12*a^2*c^2*d^8 -1686*w0^8*c^4*d^4 +15*w0^12*a^8*c^4 +180*w0^12*a^2*b^4*c^2*d^4 +356*w0^10*a^4*d^6 +692*w0^10*b^4*d^6 -6876*w0^8*a^2*b^2*d^4 +74*w0^10*a^2*d^8 -28044*w0^8*a^4*c^2*d^2 +826*w0^10*a^2*b^8 +1316*w0^10*a^6*b^4 +60*w0^12*a^4*b^2*d^6 +30*w0^12*b^8*c^2*d^2 -2346.6666667*w0^9*a^2*b^2*c^2*d^3*sqrt3 +6*w0^12*c^10*d^2 +1128*w0^10*a^6*c^2*d^2 +197048.8888889*w0^7*a^2*c^2*d^3*sqrt3 +60*w0^12*a^4*c^6*d^2 +60*w0^12*a^6*c^4*d^2 -672573.7777778*w0^6*a^2*d^4 +386*w0^10*a^8*c^2 +45052.8888889*w0^6*a^4*d^2 -13752*w0^8*a^2*b^2*c^2*d^2 +4392*w0^10*a^4*b^2*c^2*d^2 +22080576.1316872*w0^3*d^3*sqrt3 -586.6666667*w0^9*a^2*c^2*d^5*sqrt3 -1760*w0^9*a^4*c^4*d*sqrt3 -281*w0^8*c^8"
## 
## $`xyz(0, 0, 1): `
## [1] "-20384*w0^7*a*b^7 -672*w0^9*a*b^7*c^2 -672*w0^9*a*b^3*c^6 -408874.6666667*w0^6*a^3*b*d^3*sqrt3 -25685.3333333*w0^6*a^2*b^2*c^3*sqrt3 -15680*w0^7*a*b*c^6 +5280*w0^8*a^4*b^2*c*d^2*sqrt3 -84672*w0^7*a^3*b*c^2*d^2 -2235744*w0^5*a^3*b^3 +85066.6666667*w0^4*c^5*sqrt3 -1760*w0^8*a^4*b^2*c^3*sqrt3 -408874.6666667*w0^6*a*b*d^5*sqrt3 -672*w0^9*a*b*c^2*d^6 +5280*w0^8*b^2*c*d^6*sqrt3 -1008*w0^9*a^5*b*c^4 -168*w0^9*a^9*b -881440*w0^6*a^4*c*d^2*sqrt3 +14200977.7777778*w0^4*a^2*c^3*sqrt3 +198725185.185185*w0^2*c*d^2*sqrt3 -2016*w0^9*a^5*b^3*c^2 -672*w0^9*a^3*b*d^6 +5280*w0^8*b^4*c*d^4*sqrt3 -672*w0^9*a*b^7*d^2 +129696*w0^5*a*b*c^2*d^2 -1403248*w0^5*a*b^5 -51744*w0^7*a*b^3*d^4 -672*w0^9*a*b^3*d^6 +1226624*w0^6*a*b*c^4*d*sqrt3 -56448*w0^7*a*b^5*d^2 -4032*w0^9*a^3*b^3*c^2*d^2 +953216*w0^6*b^2*c*d^4*sqrt3 -170133.3333333*w0^4*c^3*d^2*sqrt3 -94080*w0^7*a^3*b^3*c^2 +958496*w0^6*b^4*c*d^2*sqrt3 -2016*w0^9*a^3*b^3*c^4 -162976*w0^5*a*b^3*c^2 -672*w0^9*a^7*b*d^2 -2016*w0^9*a^5*b^3*d^2 -42336*w0^7*a^5*b^3 +1760*w0^8*a^2*c^5*d^2*sqrt3 +293813.3333333*w0^6*a^4*c^3*sqrt3 +5280*w0^8*a^4*c*d^4*sqrt3 +4693.3333333*w0^8*c^3*d^6*sqrt3 +5280*w0^8*a^2*c*d^6*sqrt3 -672*w0^9*a^7*b*c^2 -2016*w0^9*a^3*b*c^4*d^2 +1226624*w0^6*a^3*b*c^2*d*sqrt3 -672*w0^9*a^3*b^7 -8800*w0^6*c^3*d^4*sqrt3 -37632*w0^7*a^5*b*d^2 -1760*w0^8*a^4*c^5*sqrt3 +635477.3333333*w0^6*b^2*c^3*d^2*sqrt3 -672*w0^9*a^7*b^3 -586.6666667*w0^8*a^6*c^3*sqrt3 +1760*w0^8*a^6*c*d^2*sqrt3 -586.6666667*w0^8*c^9*sqrt3 -42602933.3333333*w0^4*a^2*c*d^2*sqrt3 +1760*w0^8*b^6*c*d^2*sqrt3 -168*w0^9*a*b*c^8 -15680*w0^7*a*b*d^6 +64848*w0^5*a*b*d^4 +8644778.6666667*w0^5*c^5*d -255200*w0^4*c*d^4*sqrt3 -42336*w0^7*a^3*b*d^4 -47040*w0^7*a*b*c^2*d^4 -5280*w0^6*c*d^6*sqrt3 -3520*w0^8*a^2*b^2*c^5*sqrt3 -19762275.5555556*w0^4*a*b*d^3*sqrt3 -37632*w0^7*a^5*b*c^2 -408874.6666667*w0^6*a*b^3*d^3*sqrt3 -1008*w0^9*a*b*c^4*d^4 -964208*w0^5*a^5*b -672*w0^9*a*b*c^6*d^2 -47040*w0^7*a*b*c^4*d^2 +1158840636.44444*w0^3*a^3*b -15442435.5555556*w0^4*b^2*c^3*sqrt3 -886720*w0^6*a^2*c*d^4*sqrt3 -1008*w0^9*a^5*b*d^4 -1008*w0^9*a*b^5*c^4 -2016*w0^9*a^3*b*c^2*d^4 +77056*w0^6*a^2*b^2*c*d^2*sqrt3 -2016*w0^9*a*b^5*c^2*d^2 +8800*w0^8*b^2*c^3*d^4*sqrt3 +8800*w0^8*a^2*c^3*d^4*sqrt3 -36630222.2222222*w0^3*a*b*c^2 +295573.3333333*w0^6*a^2*c^5*sqrt3 -56448*w0^7*a*b^5*c^2 +3520*w0^8*b^4*c^3*d^2*sqrt3 +1760*w0^8*c*d^8*sqrt3 -103488*w0^7*a*b^3*c^2*d^2 -1760*w0^8*a^2*c^7*sqrt3 -2016*w0^9*a^3*b^5*d^2 -2016*w0^9*a*b^3*c^2*d^4 -586.6666667*w0^8*b^6*c^3*sqrt3 -162976*w0^5*a*b^3*d^2 +1760*w0^6*c^7*sqrt3 -319498.6666667*w0^6*b^4*c^3*sqrt3 -66241728.3950617*w0^2*c^3*sqrt3 +817749.3333333*w0^6*a*b*c^2*d^3*sqrt3 -168*w0^9*a*b^9 -28815928.8888889*w0^5*c^3*d^3 -1008*w0^9*a*b^5*d^4 -51744*w0^7*a^3*b^5 +59286826.6666667*w0^4*a*b*c^2*d*sqrt3 -672*w0^9*a^3*b*c^6 -1008*w0^9*a^5*b^5 -2030880*w0^5*a^3*b*c^2 -591146.6666667*w0^6*a^2*c^3*d^2*sqrt3 -1232101080.88889*w0^3*a*b^3 -2016*w0^9*a^3*b^3*d^4 +1760*w0^8*b^2*c^5*d^2*sqrt3 +3520*w0^8*a^4*c^3*d^2*sqrt3 +64848*w0^5*a*b*c^4 -916845740.740741*w0*a*b +1226624*w0^6*a*b^3*c^2*d*sqrt3 -2016*w0^9*a*b^3*c^4*d^2 -2016*w0^9*a^3*b^5*c^2 -317738.6666667*w0^6*b^2*c^5*sqrt3 -168*w0^9*a*b*d^8 +7040*w0^8*a^2*b^2*c^3*d^2*sqrt3 -36630222.2222222*w0^3*a*b*d^2 +3520*w0^8*c^5*d^4*sqrt3 -1760*w0^8*b^2*c^7*sqrt3 +5280*w0^8*a^2*b^4*c*d^2*sqrt3 -1760*w0^8*a^2*b^4*c^3*sqrt3 +46327306.6666667*w0^4*b^2*c*d^2*sqrt3 -51744*w0^7*a*b^3*c^4 +8644778.6666667*w0^5*c*d^5 -10976*w0^7*a^7*b -94080*w0^7*a^3*b^3*d^2 -2030880*w0^5*a^3*b*d^2 -2016*w0^9*a^5*b*c^2*d^2 +10560*w0^8*a^2*b^2*c*d^4*sqrt3 -1760*w0^8*b^4*c^5*sqrt3 -42336*w0^7*a^3*b*c^4 -1760*w0^6*c^5*d^2*sqrt3"