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# 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
``````

08/11/2017

1 day ago

51 commits
The text passage describes a package for working with semialgebraic sets in the context of multivariate polynomials, including examples of how to define algebraic varieties and semialgebraic sets, and how to check if an algebraic set is zero-dimensional and retrieve its elements.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
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3

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# SemialgebraicSets

PackageEvaluator Build Status

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
``````

08/11/2017

1 day ago

51 commits
user
The text passage describes a package for working with semialgebraic sets in the context of multivariate polynomials, including examples of how to define algebraic varieties and semialgebraic sets, and how to check if an algebraic set is zero-dimensional and retrieve its elements.

TYPE: NO_QUESTION