RegularChains[FastArithmeticTools][ReduceCoefficientsDim0] - reduce the coefficients of a polynomial w.r.t a 0-dim regular chain
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Calling Sequence
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ReduceCoefficientsDim0(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 ReduceCoefficientsDim0 returns the normal form of f w.r.t. rc in the sense of Groebner bases.
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rc is assumed to be a normalized zero-dimensional regular chain and all variables of f but the main one must be algebraic w.r.t. rc. See the subpackage ChainTools for more information about these concepts.
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The algorithm relies on the fast division trick (based on power series inversion) and FFT-based multivariate multiplication.
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Examples
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We solve a system in 3 variables and 3 unknowns
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Its triangular decomposition consists of only one regular chain
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The polynomial in x is not normalized
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Indeed its initial is not a constant in R
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We compute the inverse of the initial of px w.r.t. rc Note that the Inverse will not fail if its first argument is not invertible w.r.t. its second one; computations will split if a zero-divisor is met. This explains the non-trivial signature of the Inverse function
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We get the inverse the initial of px w.r.t. rc
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We multiply px by the inverse of its initial and reduce the product w.r.t rc. The returned polynomial is now normalized w.r.t. rc. Note that only the polynomials of rc in y and z are used during this reduction process.
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