variety: Compute a Variety

Description Usage Arguments Value Examples

View source: R/variety.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

1
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

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
## 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.

Related to variety in algstat...