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RegularChains[SuggestVariableOrder] - Suggests a variable order for decomposing a polynomial system efficiently
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Calling Sequence
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SuggestVariableOrder(sys)
SuggestVariableOrder(sys, vars)
SuggestVariableOrder(sys, 'decomposition'='cad')
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Parameters
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sys
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polynomial system
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vars
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list of set of variables
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'decomposition'='cad'
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(optional) boolean flag
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Description
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The input argument sys is a list of constraints which can be any polynomial equation, inequation or inequality. Each constraint consisting of a polynomial (with no equality or inequality sign) is interpreted as an equation.
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The output of SuggestVariableOrder(sys) is a variable list which can then be passed as argument to PolynomialRing.
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If vars is given as an input argument, the following rules apply: (1) each indeterminate not appearing in sys or vars will be treated as parameters and therefore will be present in SuggestVariableOrder(sys) with a smaller rank than any indeterminate appearing in both sys and vars. (2) In addition, if vars is given as a list, then the orders among variables appearing in both vars and sys remain unchanged.
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The command SuggestVariableOrder(sys) computes this variable list by means of combinatorial arguments only, say by comparing vertex degrees in a suitable graph. No algebraic computations are performed. Therefore, this variable order is determined heuristically and there is no guarantee of optimality.
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Options
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The decomposition option controls the targeted type of decomposition.
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If 'decomposition'='cad' is specified and vars not supplied, then the method is best appropriate for computing a cylindrical algebraic decomposition, see CylindricalAlgebraicDecompose.
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Compatibility
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The RegularChains[SuggestVariableOrder] command was introduced in Maple 16.
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Examples
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Define a set of equations.
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![sys := [v^4+4*x*u*v^2-2*y^2*v^2-4*x^2*v^2-4*y^2*u^2+4*x*y^2*u+y^4, 4*u^2-4*x*u-y^2, 4*u^2-4*x*u-y^2]](/support/helpjp/helpview.aspx?si=5375/file02699/math181.png)
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![sys := [v^4+4*x*u*v^2-2*y^2*v^2-4*x^2*v^2-4*y^2*u^2+4*x*y^2*u+y^4, 4*u^2-4*x*u-y^2, 4*u^2-4*x*u-y^2]](/support/helpjp/helpview.aspx?si=5375/file02699/math184.png)
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Compute a variable order for it.
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![[x, u, v, y]](/support/helpjp/helpview.aspx?si=5375/file02699/math193.png)
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Use the 'decomposition'='cad' option to confirm that this order is suitable for
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![lv := [x, u, v, y]](/support/helpjp/helpview.aspx?si=5375/file02699/math204.png)
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Building a polynomial ring
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Computing a cylindrical algebraic decomposition of this challenging example
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See Also
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ComprehensiveTriangularize, CylindricalAlgebraicDecompose, GeneralConstruct, LazyRealTriangularize, PartialCylindricalAlgebraicDecomposition, PolynomialRing, RealComprehensiveTriangularize, RealRootClassification, RealTriangularize, RegularChains, SamplePoints, Triangularize
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