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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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![[ChainTools, ConstructibleSetTools, Display, DisplayPolynomialRing, Equations, ExtendedRegularGcd, FastArithmeticTools, Inequations, Info, Initial, Intersect, Inverse, IsRegular, LazyRealTriangularize, MainDegree, MainVariable, MatrixCombine, MatrixTools, NormalForm, ParametricSystemTools, PolynomialRing, Rank, RealTriangularize, RegularGcd, RegularizeInitial, SamplePoints, SemiAlgebraicSetTools, Separant, SparsePseudoRemainder, SuggestVariableOrder, Tail, Triangularize]](/support/helpjp/helpview.aspx?si=5682/file06498/math109.png)
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![[BivariateModularTriangularize, IteratedResultantDim0, IteratedResultantDim1, NormalFormDim0, NormalizePolynomialDim0, NormalizeRegularChainDim0, RandomRegularChainDim0, RandomRegularChainDim1, ReduceCoefficientsDim0, RegularGcdBySpecializationCube, RegularizeDim0, ResultantBySpecializationCube, SubresultantChainSpecializationCube]](/support/helpjp/helpview.aspx?si=5682/file06498/math116.png)
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![variables := [x, y, z]](/support/helpjp/helpview.aspx?si=5682/file06498/math127.png) 
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We solve a system in 3 variables and 3 unknowns
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![lrc := [regular_chain]](/support/helpjp/helpview.aspx?si=5682/file06498/math157.png)
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Its triangular decomposition consists of only one regular chain
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![rc := lrc[1]](/support/helpjp/helpview.aspx?si=5682/file06498/math163.png)
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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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![linv := [[[174020324*z^19+197335754*z^18+7625943*z^17+198840137*z^16+378204215*z^15+815531348*z^14+358244196*z^13+680868023*z^12+248247024*z^11+563170682*z^10+678017442*z^9+232546371*z^8+493675934*z^7+717866054*z^6+661798200*z^5+439140691*z^4+372603338*z^3+113779500*z^2+110488854*z+493921163, 1, regular_chain]], []]](/support/helpjp/helpview.aspx?si=5682/file06498/math205.png)
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We get the inverse the initial of px w.r.t. rc
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![invipx := linv[1][1][1]](/support/helpjp/helpview.aspx?si=5682/file06498/math213.png)
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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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