variety: Compute a Variety

In dkahle/bertini: Interface to 'Bertini'

Compute a Variety

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

if (has_bertini()) {


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"))


#
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)

}

dkahle/bertini documentation built on July 16, 2022, 9:26 a.m.