variety: Compute a Variety

In algstat: Algebraic statistics in R

Description

The variety of a collection of multivariate polynomials is the collection of points at which those polynomials are (simultaneously) equal to 0. variety uses Bertini to find this set.

Usage

variety(mpolyList, varOrder, ...)

Arguments

mpolyList	Bertini code as either a character string or function; see examples
varOrder	variable order (see examples)
...	stuff to pass to bertini

Value

an object of class bertini

Examples

## Not run: 


polys <- mp(c(
  "x^2 - y^2 - z^2 - .5",
  "x^2 + y^2 + z^2 - 9",
  ".25 x^2 + .25 y^2 - z^2"
))
variety(polys)

# algebraic solution :
c(sqrt(19)/2, 7/(2*sqrt(5)), 3/sqrt(5)) # +/- each ordinate



# character vectors can be taken in; they're passed to mp
variety(c("y - x^2", "y - x - 2"))



# an example of how varieties are invariant to the
# the generators of the ideal
variety(c("2 x^2 + 3 y^2 - 11", "x^2 - y^2 - 3"))

# the following takes a few seconds to initialize, feel free to them
# gb <- grobner(mp(c("2 x^2 + 3 y^2 - 11", "x^2 - y^2 - 3")))
# variety(gb)

m2("
R = QQ[x,y]
gens gb ideal(2*x^2 + 3*y^2 - 11, x^2 - y^2 - 3)
")
variety(c("y^2 - 1", "x^2 - 4"))
variety(c("x^2 - 4", "y^2 - 1"))



# variable order is by default equal to vars(mpolyList)
# (this finds the zeros of y = x^2 - 1)
variety(c("y", "y - x^2 + 1")) # y, x
vars(mp(c("y", "y - x^2 + 1")))
variety(c("y", "y - x^2 + 1"), c("x", "y")) # x, y



# complex solutions
variety("x^2 + 1")
variety(c("x^2 + 1 + y", "y"))


# multiplicities
variety("x^2")
variety(c("2 x^2 + 1 + y", "y + 1"))
variety(c("x^3 - x^2 y", "y + 2"))


#
p <- mp(c("2 x  -  2  -  3 x^2 l  -  2 x l",
  "2 y  -  2  +  2 l y",
  "y^2  -  x^3  -  x^2"))
variety(p)


## End(Not run)

algstat documentation built on May 29, 2017, 10:34 p.m.