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RegularChains[FastArithmeticTools]

  

ReduceCoefficientsDim0

  

reduce the coefficients of a polynomial w.r.t a 0-dim regular chain

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

ReduceCoefficientsDim0(f, rc, R)

Parameters

R

-

a polynomial ring

rc

-

a regular chain of R

f

-

polynomial of R

Description

• 

The command ReduceCoefficientsDim0 returns the normal form of f w.r.t. rc in the sense of Groebner bases.

• 

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.

• 

R must have a prime characteristic p such that FFT-based polynomial arithmetic can be used for this computation. The higher the degrees of f and rc are, the larger must be e such that 2e divides p1.  If the degree of f or rc is too large, then an error is raised.

• 

The algorithm relies on the fast division trick (based on power series inversion) and FFT-based multivariate multiplication.

Examples

withRegularChains

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

(1)

withFastArithmeticTools

BivariateModularTriangularize,IteratedResultantDim0,IteratedResultantDim1,NormalFormDim0,NormalizePolynomialDim0,NormalizeRegularChainDim0,RandomRegularChainDim0,RandomRegularChainDim1,ReduceCoefficientsDim0,RegularGcdBySpecializationCube,RegularizeDim0,ResultantBySpecializationCube,SubresultantChainSpecializationCube

(2)

withChainTools

Chain,ChangeOfOrder,Construct,Cut,DahanSchostTransform,Dimension,Empty,EqualSaturatedIdeals,EquiprojectableDecomposition,Extend,ExtendedNormalizedGcd,IsAlgebraic,IsEmptyChain,IsInRadical,IsInSaturate,IsIncluded,IsPrimitive,IsStronglyNormalized,IsZeroDimensional,IteratedResultant,LastSubresultant,Lift,ListConstruct,NormalizeRegularChain,NumberOfSolutions,Polynomial,Regularize,RemoveRedundantComponents,SeparateSolutions,Squarefree,SquarefreeFactorization,SubresultantChain,SubresultantOfIndex,Under,Upper

(3)

variablesx,y,z;p957349889

variables:=x,y,z

p:=957349889

(4)

sysx5+y53y1,5y43,20x+yz

sys:=20x+yz,5y43,x5+y53y1

(5)

RPolynomialRingvariables,p

R:=polynomial_ring

(6)

We solve a system in 3 variables and 3 unknowns

lrcTriangularizesys,R

lrc:=regular_chain

(7)

Its triangular decomposition consists of only one regular chain

rclrc1

rc:=regular_chain

(8)

Equationsrc,R

z12+94127136z8+691135635z7+676458799z4+195425386z3+326553470z2+574327669x+27352854z13+673373922z9+410681381z8+817312291z5+308837227z4+32655347z3+116876413z+880926729,z12+94127136z8+691135635z7+676458799z4+195425386z3+326553470z2+574327669y+547057079z13+927802747z9+821042762z8+352188797z5+237219820z4+326553470z3+805850702z+386236578,z20+957349886z16+944549889z15+886639826z12+458149889z11+156173647z10+568152312z8+120112423z7+434195336z6+398220483z5+536874419z4+604689895z3+446611758z2+237311560z+665813406

(9)

The polynomial in x is not normalized

pxPolynomialx,rc,R

px:=xz12+27352854z13+94127136xz8+673373922z9+691135635xz7+410681381z8+676458799xz4+817312291z5+195425386xz3+308837227z4+326553470xz2+32655347z3+574327669x+116876413z+880926729

(10)

Indeed its initial is not a constant in R

ipxInitialpx,R

ipx:=z12+94127136z8+691135635z7+676458799z4+195425386z3+326553470z2+574327669

(11)

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

linvInverseipx,rc,R

linv:=174020324z19+197335754z18+7625943z17+198840137z16+378204215z15+815531348z14+358244196z13+680868023z12+248247024z11+563170682z10+678017442z9+232546371z8+493675934z7+717866054z6+661798200z5+439140691z4+372603338z3+113779500z2+110488854z+493921163,1,regular_chain,

(12)

We get the inverse the initial of px w.r.t. rc 

invipxlinv111

invipx:=174020324z19+197335754z18+7625943z17+198840137z16+378204215z15+815531348z14+358244196z13+680868023z12+248247024z11+563170682z10+678017442z9+232546371z8+493675934z7+717866054z6+661798200z5+439140691z4+372603338z3+113779500z2+110488854z+493921163

(13)

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.

ReduceCoefficientsDim0invipxpx,rc,R

703897958z19+637307906z18+745731651z17+21899044z16+658962013z15+899050902z14+77671904z13+921629286z12+870449919z11+122035854z10+791154398z9+547395190z8+624024465z7+710904034z6+4427709z5+954705258z4+221310023z3+584706443z2+x+332923317z+743851316

(14)

See Also

ChainTools

NormalForm

NormalFormDim0

NormalizePolynomialDim0

NormalizeRegularChainDim0

RegularChains

 


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