Re: PolynomialMod

*To*: mathgroup at smc.vnet.net*Subject*: [mg121357] Re: PolynomialMod*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Mon, 12 Sep 2011 04:21:03 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201109111128.HAA12187@smc.vnet.net> <52BDE91D-1ACE-471D-A992-DF8902ADBE7D@mimuw.edu.pl> <sig.0235ae8fc1.4E6CFABB.9040206@csl.pl> <6763942A-FD6A-4C87-AB48-51BF0CC71021@mimuw.edu.pl>

In fact, for univariate polynomials you can equally well use = PolynomialRemaineder: PolynomialRemainder[1+3 x+5 x^2+5 x^3+5 x^4+3 x^5+x^6,x^3-b x-c,x] x^2 (b^2+5 b+3 c+5)+x (3 b^2+2 b c+5 b+5 c+3)+3 b c+c^2+5 c+1 Andrzej On 11 Sep 2011, at 21:38, Andrzej Kozlowski wrote: > > On 11 Sep 2011, at 20:15, Artur wrote: > >> Mayby I was used wrong example >> >> Example 1. >> f = 1 + 3 x + 5 x2 + 5 x3 + 5 x4 + 3 x5 + x6; >> >> If we have smaller order equation for example >> x^5-x-1=0 >> that x^5=x+1 >> multiply both sides by x >> x^6=x^2+x >> if we substitute in f x^6->x^2+x and x^5->x+1 we have >> f = 1 + 3 x + 5 x2 + 5 x3 + 5 x4 >> + 3 (x+1) +(x^2+x); >> now after reduction f=4 + 7 x + 6 x^2 + 5 x^3 + 5 x^4 >> >> That same result we will obtain uses >> >> PolynomialMod[1 + 3 x + 5 x^2 + 5 x^3 + 5 x^4 + 3 x^5 + x^6, x^5 - x = - 1] >> >> of course f=0 is divisible by my p from previous example but we can = do similar cycle of substitutions as in Example 1 but if f<>0 isn't = divisible >> >> in my example let symbolically >> >> x^3-b x-c==0 >> >> x^3=b x+c >> x^4=b x^2+c x >> x^5=b x^3+c x^2=b (b x+c)+c x^2=b^2 x+b c+c x^2 >> x^6=b^2 x^2+b c x+c x^3=b^2 x^2+b c x+c (b x+c)=b^2 x^2+2 b c = x+c^2 >> and apply these equations to starting function f >> >> f=1 + 3 x + 5 x >> 2 >> + 5 (b x+c) + 5 (b x^2+c x) + 3 (b^2 x+b c+c x^2) + (b^2 x^2+2 b c = x+c^2) >> after reduction we have finally result (Collect by x) >> >> >> f=(1 + 5 c + 3 b c + c^2) + (3 + 5 b + 3 b^2 + 5 c + 2 b c) x + (5 = + 5 b + b^2 + 3 c) x^2 >> >> generally g=0 and f is some function (not necessary zero) >> >> Artur Jasinski >> > > > I think what you want to do is to use PolynomialReduce and not = PolynomialMod. In your case > > In[11]:= Last[ > PolynomialReduce[1 + 3*x + 5*x^2 + 5*x^3 + 5*x^4 + 3*x^5 + x^6, > x^3 - b*x - c, x]] > > Out[11]= x^2*(b^2 + 5*b + 3*c + 5) + > x*(3*b^2 + 2*b*c + 5*b + 5*c + 3) + > 3*b*c + c^2 + 5*c + 1 > > > > Andrzej Kozlowski >

**References**:**PolynomialMod***From:*Artur <grafix@csl.pl>