groebner {qspray}

Gröbner basis

Description

Returns a Gröbner basis following Buchberger's algorithm using the lexicographical order.

Usage

groebner(G, minimal = TRUE, reduced = TRUE)

Arguments

G

a list of qspray polynomials, the generators of the ideal

minimal

Boolean, whether to return a minimal basis

reduced

Boolean, whether to return the reduced basis

Value

A Gröbner basis of the ideal generated by G, given as a list of qspray polynomials.

References

Cox, Little & O'Shea. Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra. Fourth edition, Springer 2015.

Examples

library(qspray)
f <- qsprayMaker(string = "x^(3) - 2 x^(1,1)")
g <- qsprayMaker(string = "x^(2,1) - 2 x^(0,2) + x^(1)")
groebner(list(f, g), FALSE, FALSE)
# other example
x <- qlone(1); y <- qlone(2); z <- qlone(3)
f1 <- x^2 + y + z^2 - 1
f2 <- x^2 + y + z - 1
f3 <- x + y^2 + z - 1
gb <- groebner(list(f1, f2, f3))
lapply(gb, prettyQspray, vars = c("x", "y", "z"))