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)
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(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)
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(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 DecompositionExplore with Wolfram|Alpha
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References
Strzebonski, A. "Solving Algebraic Inequalities." Mathematica J. 7, 525-541, 2000.Referenced on Wolfram|Alpha
Cylindrical PartsCite this as:
Weisstein, Eric W. "Cylindrical Parts." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CylindricalParts.html