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AIrreduc

inert absolute irreducibility function 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

 > $f≔92{x}^{9}y-93{x}^{2}{y}^{2}+91{x}^{7}{y}^{3}+{y}^{4}+{x}^{10}$
 ${f}{≔}{{x}}^{{10}}{+}{92}{}{{x}}^{{9}}{}{y}{+}{91}{}{{x}}^{{7}}{}{{y}}^{{3}}{-}{93}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}$ (1)
 > $\mathrm{evala}\left(\mathrm{AIrreduc}\left(f\right)\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{alias}\left(a=\mathrm{RootOf}\left({T}^{3}-2\right)\right)$
 ${a}$ (3)
 > $f≔a{y}^{3}+a{y}^{3}x+13a{y}^{2}-7xy-7{x}^{2}y-91x$
 ${f}{≔}{a}{}{{y}}^{{3}}{}{x}{+}{a}{}{{y}}^{{3}}{+}{13}{}{a}{}{{y}}^{{2}}{-}{7}{}{{x}}^{{2}}{}{y}{-}{7}{}{x}{}{y}{-}{91}{}{x}$ (4)
 > $\mathrm{evala}\left(\mathrm{AIrreduc}\left(f\right)\right)$
 ${\mathrm{false}}$ (5)
 > $\mathrm{evala}\left(\mathrm{AFactor}\left(f\right)\right)$
 ${-}{7}{}\left({-}\frac{{a}{}{{y}}^{{2}}}{{7}}{+}{x}\right){}\left({x}{}{y}{+}{y}{+}{13}\right)$ (6)
 > $f≔-2{y}^{2}{x}^{2}+{y}^{2}+10{x}^{2}y+25{x}^{4}$
 ${f}{≔}{25}{}{{x}}^{{4}}{-}{2}{}{{y}}^{{2}}{}{{x}}^{{2}}{+}{10}{}{{x}}^{{2}}{}{y}{+}{{y}}^{{2}}$ (7)
 > $\mathrm{evala}\left(\mathrm{AIrreduc}\left(f\right)\right)$
 ${\mathrm{FAIL}}$ (8)
 > $\mathrm{evala}\left(\mathrm{AFactor}\left(f\right)\right)$
 ${25}{}\left(\frac{{x}{}{y}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{2}\right)}{{5}}{+}{{x}}^{{2}}{+}\frac{{y}}{{5}}\right){}\left({-}\frac{{x}{}{y}{}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{2}\right)}{{5}}{+}{{x}}^{{2}}{+}\frac{{y}}{{5}}\right)$ (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.

 > $f≔48778{x}^{3}+275894451{y}^{6}-1188761805{y}^{5}x+1707361425{y}^{4}{x}^{2}-817400375{x}^{3}{y}^{3}+1232777{y}^{3}$
 ${f}{≔}{-}{817400375}{}{{x}}^{{3}}{}{{y}}^{{3}}{+}{1707361425}{}{{y}}^{{4}}{}{{x}}^{{2}}{-}{1188761805}{}{{y}}^{{5}}{}{x}{+}{275894451}{}{{y}}^{{6}}{+}{48778}{}{{x}}^{{3}}{+}{1232777}{}{{y}}^{{3}}$ (10)
 > $\mathrm{evala}\left(\mathrm{AIrreduc}\left(f\right)\right)$
 ${\mathrm{FAIL}}$ (11)
 > $\mathrm{evala}\left(\mathrm{AIrreduc}\left(f,\left\{37,43\right\}\right)\right)$
 ${\mathrm{true}}$ (12) References

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