## SemialgebraicSets.jl

Extension of MultivariatePolynomials to semialgebraic sets
Popularity
15 Stars
Updated Last
1 Year Ago
Started In
August 2017

# SemialgebraicSets

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```