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RegularChains[FastArithmeticTools]

  

NormalizeRegularChainDim0

  

normalize a zero-dimensional regular chain

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

NormalizeRegularChainDim0(rc, R)

Parameters

R

-

polynomial ring

rc

-

a regular chain of R

Description

• 

Returns a normalized regular chain generating the same ideal as rc.

• 

rc is a zero-dimensional non-empty regular chain.

• 

Moreover R must have a prime characteristic p such that FFT-based polynomial arithmetic can be used for this actual 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.

Examples

withRegularChains:

withFastArithmeticTools:

withChainTools:

variablesx,y,z:p957349889:

sys5y43,20x+yz,x5+y53y1:

RPolynomialRingvariables,p

Rpolynomial_ring

(1)

We solve a system in 3 variables and 3 unknowns

lrcTriangularizesys,R

lrcregular_chain

(2)

Its triangular decomposition consists of only one regular chain

rclrc1

rcregular_chain

(3)

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

(4)

Each initial is not equal to 1, hence this regular chain is not normalized

mapInitial,Equationsrc,R,R

z12+94127136z8+691135635z7+676458799z4+195425386z3+326553470z2+574327669,z12+94127136z8+691135635z7+676458799z4+195425386z3+326553470z2+574327669,1

(5)

We compute here a regular chain which is normalized and which describes the same solution as the previous one

nrcNormalizeRegularChainDim0rc,R

nrcregular_chain

(6)

We check that it is normalized

Equationsnrc,R:mapInitial,Equationsnrc,R,R

1,1,1

(7)

We check that the two regular chains describe the set of solutions

EqualSaturatedIdealsrc,nrc,R

true

(8)

See Also

NormalForm

NormalFormDim0

NormalizePolynomialDim0

ReduceCoefficientsDim0

RegularChains