RegularChains[FastArithmeticTools][NormalizePolynomialDim0] - normalize a polynomial w.r.t a 0-dim regular chain
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Calling Sequence
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NormalizePolynomialDim0(f, rc, R)
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Parameters
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R
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a polynomial ring
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rc
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a regular chain of R
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f
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polynomial of R
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Description
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The command NormalizePolynomialDim0 returns a normalized form of f w.r.t. rc, that is, a polynomial which is associated to f modulo rc, such that is normalized w.r.t. rc.
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rc is zero-dimensional regular chain, and f together with rc forms a zero-dimensional regular chain.
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Examples
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We consider two bivariate polynomials and want to compute their common solutions
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We first compute their subresultant chain using FFT techniques
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We deduce their resultants
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We observe below that no root of r2 cancels the leading coefficients of f1 or f2. Hence, any roots of r2 can be extended into a solution of the system by a GCD computation.
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We define the regular chain consisting of r2
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We compute the GCD of f1 and f2 modulo r2
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![g2 := [[x*y^2+x*y+2*x+y^2+y+2+x^3, regular_chain], [x*y^2+x*y+2*x+y^2+y+2+x^3, regular_chain]]](/support/helpjp/helpview.aspx?si=6448/file06494/math187.png)
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We normalize this GCD w.r.t. r2 which leads to a simpler expression with one as leading coefficient
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