Homogenize a polynomial

Homogenize a polynomial.

homogenize(x, indeterminate = "t")

dehomogenize(x, indeterminate = "t")

is.homogeneous(x)

homogeneous_components(x)

Arguments

x	an mpoly object, see mpoly()
indeterminate	name of homogenization

Value

a (de/homogenized) mpoly or an mpolyList

Examples

x <- mp("x^4 + y + 2 x y^2 - 3 z")
is.homogeneous(x)
#> [1] FALSE
(xh <- homogenize(x))
#> x^4  +  y t^3  +  2 x y^2 t  -  3 z t^3
is.homogeneous(xh)
#> [1] TRUE

homogeneous_components(x)
#> y  -  3 z
#> 2 y^2 x
#> x^4

homogenize(x, "o")
#> x^4  +  y o^3  +  2 x y^2 o  -  3 z o^3

xh <- homogenize(x)
dehomogenize(xh) # assumes indeterminate = "t"
#> x^4  +  y  +  2 x y^2  -  3 z
plug(xh, "t", 1) # same effect, but dehomogenize is faster
#> x^4  +  y  +  2 x y^2  -  3 z



# the functions are vectorized
(ps <- mp(c("x + y^2", "x + y^3")))
#> x  +  y^2
#> x  +  y^3
(psh <- homogenize(ps))
#> x t  +  y^2
#> x t^2  +  y^3
dehomogenize(psh)
#> x  +  y^2
#> x  +  y^3


# demonstrating a leading property of homogeneous polynomials
library(magrittr)
#> 
#> Attaching package: ‘magrittr’
#> The following object is masked from ‘package:tidyr’:
#> 
#>     extract
p  <- mp("x^2 + 2 x + 3")
(ph <- homogenize(p, "y"))
#> x^2  +  2 x y  +  3 y^2
lambda <- 3
(d <- totaldeg(p))
#> [1] 2
ph %>%
  plug("x", lambda*mp("x")) %>%
  plug("y", lambda*mp("y"))
#> 9 x^2  +  18 x y  +  27 y^2
lambda^d * ph
#> 9 x^2  +  18 x y  +  27 y^2