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AIrreduc

inert absolute irreducibility function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

AIrreduc(P)

AIrreduc(P, S)

Parameters

P

-

multivariate polynomial

S

-

(optional) set or list of prime integers

Description

• 

The AIrreduc function is a placeholder for testing the absolute irreducibility of the polynomial P, that is irreducibility over an algebraic closure of its coefficient field. It is used in conjunction with evala.

• 

The call evala(AIrreduc(P)) tests the absolute irreducibility of the polynomial P over the field of complex numbers. The polynomial P must have algebraic number coefficients in RootOf notation (see algnum).

• 

A univariate polynomial is absolutely irreducible if and only if it is of degree 1.

• 

The function AIrreduc looks for sufficient conditions of absolute reducibility or irreducibility. It returns true if the polynomial P is detected absolutely irreducible, false if it is detected absolutely reducible, FAIL otherwise.

• 

In the case of nonrational coefficients, only trivial conditions are tested.

• 

If the polynomial P has rational coefficients, an absolute irreducibility criterion is sought over the reduction of P modulo p, where p runs through a set of prime integers. If S is given, the primes in S are used. Otherwise, the first ten odd primes and the first five primes greater than the degree of P are chosen. Although the probability for P to be absolutely reducible in case of failure is not controlled, it is very likely that P can be factored.

Examples

f92x9y93x2y2+91x7y3+y4+x10

fx10+92x9y+91x7y393x2y2+y4

(1)

evalaAIrreducf

true

(2)

aliasa=RootOfT32

a

(3)

fay3+ay3x+13ay27xy7x2y91x

fay3x+ay3+13ay27x2y7xy91x

(4)

evalaAIrreducf

false

(5)

evalaAFactorf

7xy+y+1317ay2+x

(6)

f2y2x2+y2+10x2y+25x4

f25x42x2y2+10x2y+y2

(7)

evalaAIrreducf

FAIL

(8)

evalaAFactorf

2515xyRootOf_Z22+x2+15y15xyRootOf_Z22+x2+15y

(9)

The following polynomial is absolutely irreducible, but has been specially constructed to deceive the test. This example illustrates the usefulness of the optional argument.

f48778x3+275894451y61188761805y5x+1707361425y4x2817400375x3y3+1232777y3

f817400375x3y3+1707361425x2y41188761805xy5+275894451y6+48778x3+1232777y3

(10)

evalaAIrreducf

FAIL

(11)

evalaAIrreducf,37,43

true

(12)

References

  

Ragot, Jean-Francois. "Probabilistic Absolute Irreducibility Test of Polynomials." In Proceedings of MEGA '98. 1998.

See Also

AFactor

alias

evala

irreduc

Irreduc

RootOf

 


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