TOPICS
Search

Cylindrical Parts


The cylindrical parts of a system of real algebraic equations and inequalities in variables {x_1,...,x_n} are the terms

f_1<=x_1<=g_1
(1)
f_2(x_1)<=x_2<=g_2(x_1)
(2)
|
(3)
f_n(x_1,x_2,...,x_n)<=x_n<=g_n(x_1,...,x_(n-1)),
(4)

where '<=' is one of <, <=, or =, and f_i and g_i are +/-infty or algebraic expressions in variables {x_1,...,x_(i-1)} that are real-valued for all (i-1)-tuples of real numbers {a_1,...,a_(i-1)} satisfying

f_1<=a_1<=g_1
(5)
f_2(a_1)<=a_2<=g_2(a_2)
(6)
|
(7)
f_(i-1)(a_1,...,a_(i-2))<=a_(i-1)<=g_(i-1)(a_1,...,a_(i-2)).
(8)

The conjunction of a finite number of disjoint cylindrical parts is called a cylindrical algebraic decomposition.


See also

Cylindrical Algebraic Decomposition

Explore with Wolfram|Alpha

References

Strzebonski, A. "Solving Algebraic Inequalities." Mathematica J. 7, 525-541, 2000.

Referenced on Wolfram|Alpha

Cylindrical Parts

Cite this as:

Weisstein, Eric W. "Cylindrical Parts." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CylindricalParts.html

Subject classifications