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RegularChains[ChainTools][RemoveRedundantComponents] - remove redundant quasi-components from a list of regular chains
RegularChains[SemiAlgebraicSetTools][RemoveRedundantComponents] - remove redundant quasi-components from a list of regular semi-algebraic systems
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Calling Sequence
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RemoveRedundantComponents(lrc, R)
RemoveRedundantComponents(lrsas, R)
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Parameters
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lrc
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list of regular chains
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lrsas
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list of regular semi-algebraic systems
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R
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polynomial ring
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Description
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The command RemoveRedundantComponents(lrsas, R) returns a list  of regular semi-algebraic system whose zero sets are pairwise noninclusive, and such that lrsas and  have the same zero set.
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For more details, see Algorithm 35 in the Ph.D. thesis of Yuzhen Xie.
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Compatibility
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The RegularChains[SemiAlgebraicSetTools][RemoveRedundantComponents] command was introduced in Maple 16.
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The lrsas parameter was introduced in Maple 16.
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Examples
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Consider a polynomial ring with two variables
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![R := PolynomialRing([y, x])](/support/helpjp/helpview.aspx?si=6384/file06460/math121.png)
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Consider two regular chains in R
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![rc1 := Chain([y*(y+1)], Empty(R), R)](/support/helpjp/helpview.aspx?si=6384/file06460/math130.png)
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![rc2 := Chain([x, y], Empty(R), R)](/support/helpjp/helpview.aspx?si=6384/file06460/math137.png)
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The solutions of one are contained in those of the other. The redundant one will be removed as follows
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![out := RemoveRedundantComponents([rc1, rc2], R)](/support/helpjp/helpview.aspx?si=6384/file06460/math146.png)
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![out := [regular_chain]](/support/helpjp/helpview.aspx?si=6384/file06460/math149.png)
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![[[y^2+y]]](/support/helpjp/helpview.aspx?si=6384/file06460/math156.png)
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The case of semi-algebraic system.
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![C1 := [0 < a, 0 < b, 0 < c, a < b+c, b < a+c, c < a+b, b^2+a^2-c^2 <= 0]](/support/helpjp/helpview.aspx?si=6384/file06460/math162.png)
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![C2 := [0 < a, 0 < b, 0 < c, a < b+c, b < a+c, c < a+b, c*(b^2+a^2-c^2)^2 < a*b^2*(2*a*c-c^2-a^2+b^2)]](/support/helpjp/helpview.aspx?si=6384/file06460/math166.png)
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![C3 := [a-c < 0, 0 < a, 0 < b, 0 < c, a < b+c, b < a+c, c < a+b]](/support/helpjp/helpview.aspx?si=6384/file06460/math170.png)
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![S := [C1, C2, C3]](/support/helpjp/helpview.aspx?si=6384/file06460/math174.png)
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![S := [[0 < a, 0 < b, 0 < c, a < b+c, b < a+c, c < a+b, b^2+a^2-c^2 <= 0], [0 < a, 0 < b, 0 < c, a < b+c, b < a+c, c < a+b, c*(b^2+a^2-c^2)^2 < a*b^2*(2*a*c-c^2-a^2+b^2)], [a-c < 0, 0 < a, 0 < b, 0 < c, a < b+c, b < a+c, c < a+b]]](/support/helpjp/helpview.aspx?si=6384/file06460/math177.png)
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![R := PolynomialRing([a, b, c])](/support/helpjp/helpview.aspx?si=6384/file06460/math181.png)
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![dec1 := [regular_semi_algebraic_system, regular_semi_algebraic_system, regular_semi_algebraic_system, regular_semi_algebraic_system]](/support/helpjp/helpview.aspx?si=6384/file06460/math188.png)
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![dec2 := [regular_semi_algebraic_system]](/support/helpjp/helpview.aspx?si=6384/file06460/math195.png)
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References
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Xie, Y. "Fast Algorithms, Modular Methods, Parallel Approaches and Software Engineering for Solving Polynomial Systems Symbolically" Ph.D. Thesis, University of Western Ontario, Canada, 2007.
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