SemialgebraicSets.jl

Extension of MultivariatePolynomials to semialgebraic sets
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August 2017

SemialgebraicSets

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Extension of MultivariatePolynomials to semialgebraic sets, i.e. sets defined by inequalities and equalities between polynomials. The following example shows how to build an algebraic set/algebraic variety

using TypedPolynomials
@polyvar x y z
# Algebraic variety https://en.wikipedia.org/wiki/Algebraic_variety#/media/File:Elliptic_curve2.png
@set y^2 == x^3 - x
@set x^3 == 2x*y && x^2*y == 2y^2 - x
@set x*y^2 == x*z - y && x*y == z^2 && x == y*z^4
@set x^4*y^2 == z^5 && x^3*y^3 == 1 && x^2*y^4 == 2z
@set x == z^2 && y == z^3

Once the algebraic set has been created, you can check whether it is zero-dimensional and if it is the case, you can get the finite number of elements of the set simply by iterating over it, or by using collect to transform it into an array containing the solutions.

V = @set y == x^2 && z == x^3
iszerodimensional(V) # should return false
V = @set x^2 + x == 6 && y == x+1
iszerodimensional(V) # should return true
collect(V) # should return [[2, 3], [-3, -2]]

The following example shows how to build an basic semialgebraic set

using TypedPolynomials
@polyvar x y
@set x^2 + y^2 <= 1 # Euclidean ball
# Cutting the algebraic variety https://en.wikipedia.org/wiki/Algebraic_variety#/media/File:Elliptic_curve2.png
@set y^2 == x^3 - x && x <= 0
@set y^2 == x^3 - x && x >= 1