SemialgebraicComponentInstances
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SemialgebraicComponentInstances
SemialgebraicComponentInstances[ineqs,{x1,x2,…}]
gives at least one sample point in each connected component of the semialgebraic set defined by the inequalities ineqs in the variables x1, x2, ….
Details
- SemialgebraicComponentInstances assumes that all variables are real.
- Lists or logical combinations of inequalities can be given.
- Any solution to the set of inequalities can be connected by a continuous path to one of the points returned by SemialgebraicComponentInstances.
- SemialgebraicComponentInstances produces a list of rules for variables, of the same type as Solve.
Examples
open allclose allBasic Examples (1)Summary of the most common use cases
Scope (3)Survey of the scope of standard use cases
A univariate polynomial inequality:
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-kp30q2
Multivariate polynomial equations and inequalities:
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-i2tytl
Boolean combinations of equations and inequalities:
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-crqhtd
Applications (4)Sample problems that can be solved with this function
Find at least one point in each interval defined by a univariate polynomial inequality:
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-hae6rp
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-dv0i46
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-cald64
With a weak inequality you also get the roots:
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-cpllh4
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-ihegys
Find at least one point in each connected component of a two-dimensional planar set:
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-ivqkf1
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-l2r
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-vwb
Find at least one point in each connected component of a surface:
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-d0h24d
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-6zzro
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-crf0q
Find at least one point in each connected component of a solid:
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-ru1xd
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-g68zk6
The points satisfy the inequalities:
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-n9i7w
Use the points to check whether a numerically obtained graphic is missing parts of the set:
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-qtp57b
Properties & Relations (2)Properties of the function, and connections to other functions
The returned instances satisfy the input inequalities:
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-flwnxa
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-fn0wdx
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-l567x
Use FindInstance to find a single instance satisfying the inequalities:
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-xz7sc
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-732uw
Use CylindricalDecomposition or Reduce to get a full description of the solution set:
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-iyqyig
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-in5o3
An empty list is returned if the inequalities have no solutions:
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-cht9ue
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-e19
An equivalent result can be obtained using Resolve:
https://wolfram.com/xid/0rbwdczlafwv3qzwwkw3p-o2e
Wolfram Research (2007), SemialgebraicComponentInstances, Wolfram Language function, https://reference.wolfram.com/language/ref/SemialgebraicComponentInstances.html.
Text
Wolfram Research (2007), SemialgebraicComponentInstances, Wolfram Language function, https://reference.wolfram.com/language/ref/SemialgebraicComponentInstances.html.
Wolfram Research (2007), SemialgebraicComponentInstances, Wolfram Language function, https://reference.wolfram.com/language/ref/SemialgebraicComponentInstances.html.
CMS
Wolfram Language. 2007. "SemialgebraicComponentInstances." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SemialgebraicComponentInstances.html.
Wolfram Language. 2007. "SemialgebraicComponentInstances." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SemialgebraicComponentInstances.html.
APA
Wolfram Language. (2007). SemialgebraicComponentInstances. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SemialgebraicComponentInstances.html
Wolfram Language. (2007). SemialgebraicComponentInstances. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SemialgebraicComponentInstances.html
BibTeX
@misc{reference.wolfram_2024_semialgebraiccomponentinstances, author="Wolfram Research", title="{SemialgebraicComponentInstances}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SemialgebraicComponentInstances.html}", note=[Accessed: 09-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_semialgebraiccomponentinstances, organization={Wolfram Research}, title={SemialgebraicComponentInstances}, year={2007}, url={https://reference.wolfram.com/language/ref/SemialgebraicComponentInstances.html}, note=[Accessed: 09-January-2025
]}