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Quotient Polynomial Rings by Maple

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Quotient Polynomial Rings by Maple

Kahtan H. Alzubaidy

Department of Mathematics, Faculty of Science, University of Benghazi

E-mail: kahtanalzubaidy@yahoo.com

Key Words: Quotient Rings, Multivariate Polynomials, Groebner Basis

Introduction.

Quotient polynomial rings over the infinite field containing < are involved. The computations concern the univariate polynomial rings and the multivariate polynomial rings in two variables. In both cases a vector space basis for the quotient is constructed. The case of of several variables includes cases of infinite dimensions. Procedure II-2 recognizes the infinite dimensional cases , however we shall restrict ourselves to the finite computations. Ring operations of addition and multiplication on the quotients are computed as well.

Goebner basis is used and the computations are carried out in Maple 13.

I) Univariate Polynomials.

Let F be a field. The polynomialring F[x] is a principal ideal domain. If I is an ideal of F[x], then I=<f(x)> for some f(x)2F[x].

We have the quotient polynomial ring F[x]/I.

F[x]/I={r(x): r(x) is the remainder of any polynomial in F[x] when it is divided by f(x)}.

Theorem 1.

If the degree of f(x) is n, then F[x]/I is an n-dimensional vector space over F with basis {1,x,..,}.

The ring operations of F[x]/I are given as follows

1) addition

F[x]/I

2) multiplication

"r[1](x)r[2](x) is the remainder of r[1](x)*r[2](x) w.r.t division by f(x). Thus r[1](x)r[2](x) in F[x]/I ."

Theorem 2.

F[x]/I is a field iff f(x) is irreducible over F.

The inverse of r(x) in the field F[x]/I is given as follows

gcd(f(x),r(x))=u(x)f(x)+v(x)r(x)=1 for some u(x), v(x)2F[x]. Therefore v(x)r(x)=1 in F[x]/I. Thus inverse(r(x))=v(x).

Proceduers

Example

(1)

$"` &Qopf;[x]/I ={ ax`^(2)+bx+c :a,b,c in `&Qopf;`, x^(3)+x+1=0}."$

(2)

>	ads:=proc(A,B,f);

>	sort(collect(A+B,x),x);

end proc;

(3)

(4)

>	mus:=proc(A,B,f) local p;

>	p:=collect(expand(A*B),x);

>	sort(rem(p,f,x),x);end proc;

proc (A, B, f) local p; p := collect(expand(A*B), x); sort(rem(p, f, x), x) end proc

(5)

(a[1]*c[2]+b[1]*b[2]+c[1]*a[2]-a[1]*a[2])*x^2+(b[1]*c[2]+c[1]*b[2]-a[1]*a[2]-a[1]*b[2]-b[1]*a[2])*x+c[1]*c[2]-a[1]*b[2]-b[1]*a[2]

(6)

(7)

(8)

(9)

>	inv:=proc(A,f);

>	gcdex(A,f,x,'u','v');

>	sort(u,x);

end proc;

(10)

(a^2-a*c+b^2)*x^2/(a^3+a^2*c-a^2*b-2*a*c^2+3*a*c*b+b^2*c+c^3-b^3)-(a^2+b*c)*x/(a^3+a^2*c-a^2*b-2*a*c^2+3*a*c*b+b^2*c+c^3-b^3)+(a^2+a*b+b^2-2*a*c+c^2)/(a^3+a^2*c-a^2*b-2*a*c^2+3*a*c*b+b^2*c+c^3-b^3)

(11)

(12)

" F[x[1],x[2],...,x[n]] is apolynomial ring over a field F. Suppose I is an ideal of F[x[1],x[2],...,x[n]]."

" Therefore I is finitely generated. i.e I=< f[1],f[2],...,f[m]> where f[i] in `F[`x[1],x[2],...,x[n]]"
.

" Therefore <LT(I)> = <(LT(g[1]),...,LT(g[t]))>= union <LT(g[i]) >."

Theorem 3.

$" F[x[1],x[2],...,x[n]]/I &cong;span{x^(alpha):x^(alpha)∉<LT(I)>}, where alpha=(alpha[1],...,alpha[n]) and x^(alpha)=( x[1]^(alpha[1]),...,x[n]^(alpha[n]) ) ."$

Corollory 4.

" F[x[1],x[2],...,x[n]]/I &cong; intersect complement(<LT(g[i])>) , as a vector space."

" Now, we restrict ourselves to the finite dimensional case of two variables x an y and a field F containing `&Qopf;`."

$"` complement(<x`^(u)>)={1,x,...,x^(u-1)}*{1,y,y^(2),...}, "$

$complement(`<,>`(y^v)) = `&x`({1, x, x^2, () .. ()}, {1, y, y^(v-1), () .. ()})$ , and

$complement(`<,>`(x^u*y^v)) = `&x`({1, x, x^(u-1), () .. ()}, {1, y, y^(v-1), () .. ()})$ .Thus we have

Theorem 5.

" F[x,y]/I &cong; intersect complements(<leading monomials >)" .

Corollory 6.

"F[x[1],x[2],...,x[n]]/I is a finitedimensional vector space over F iff there are at least two monomials of the forms x^(u) and y^(v)."

Proceduers

Procedure II.1

Leading Monomials

>	lm:=proc() local L,G,h,hh,H;

>	L:= [seq(args[i],i=1..nargs)];

>	G:=Basis(L,tdeg(x,y));

>	h:=LeadingMonomial(G,tdeg(x,y));if nops(h)=1 then h; else

>	hh:=sum(h[j],j=1..nops(h));

>	H:=sort(hh,order=plex(x,y));

>	[seq(op(s,H),s=1..nops(H))];fi

end proc;

proc () local L, G, h, hh, H; L := [seq(args[i], i = 1 .. nargs)]; G := Groebner:-Basis(L, tdeg(x, y)); h := Groebner:-LeadingMonomial(G, tdeg(x, y)); if nops(h) = 1 then h else hh := sum(h[j], j = 1 .. nops(h)); H := sort(hh, order = plex(x, y)); [seq(op(s, H), s = 1 .. nops(H))] end if end proc

(13)

(14)

(15)

(16)

(17)

(18)

Procedure II.2

>	fin:=proc() local K;

>	K:=lm(seq(args[i],i=1..nargs));

>	if gcd(K[1],K[nops(K)])=1 then print("quotient is finite dimentional")else print("quotient is infinite dimensional") fi;

end proc;

proc () local K; K := lm(seq(args[i], i = 1 .. nargs)); if gcd(K[1], K[nops(K)]) = 1 then print("quotient is finite dimentional") else print("quotient is infinite dimensional") end if end proc

(19)

(20)

(21)

(22)

(23)

(24)

Procedure II.3

>	tm:=proc() local K,p,q;

>	K:=lm(seq(args[i],i=1..nargs));

>	q:=expand(sum((op(1,K[1]))^i,i=0..op(2,K[1])-1)* sum((op(1,K[nops(K)]))^j,j=0..op(2,K[nops(K)])-1));

>	{seq(op(r, q), r = 1 .. nops(q))};

end proc;

$proc () local K, p, q; K := lm(seq(args[i], i = 1 .. nargs)); q := expand((sum(op(1, K[1])^i, i = 0 .. op(2, K[1])-1))*(sum(op(1, K[nops(K)])^j, j = 0 .. op(2, K[nops(K)])-1))); {seq(op(r, q), r = 1 .. nops(q))} end proc$

(25)

(26)

${1, x, y, x^2, x^3, y^2, y^3, x*y, x*y^2, x*y^3, x^2*y, x^2*y^2, x^2*y^3, x^3*y, x^3*y^2, x^3*y^3}$

(27)

(28)

${1, x, y, y^2, y^3, x*y, x*y^2, x*y^3}$

(29)

Procedure II.4

>	rm:=proc(w,A) local bn;

>	bn:=u->is(divide(u,w));

>	remove(bn,A);

end proc;

(30)

Procedure II.5

>	vsb:=proc() local LM,B,A,w;

>	LM:=lm(seq(args[i],i=1..nargs));

>	B:={seq(LM[t],t=2..nops(LM)-1)};A:=tm(seq(args[i],i=1..nargs));

>	for w in B do A:=rm(w,A) od: A;

end proc;

proc () local LM, B, A, w; LM := lm(seq(args[i], i = 1 .. nargs)); B := {seq(LM[t], t = 2 .. nops(LM)-1)}; A := tm(seq(args[i], i = 1 .. nargs)); for w in B do A := rm(w, A) end do; A end proc

(31)

(32)

${1, x, y, x^2, x^3, y^2, y^3, x*y, x*y^2, x^2*y, x^2*y^2, x^3*y}$

(33)

(34)

${1, x, y, y^2, y^3, x*y}$

(35)

Ring Operations

Addition

Procedure II.6

>	adm:=proc(A,B,L) local G;

>	G:=Basis(L,tdeg(x,y)); NormalForm(A+B,G,tdeg(x,y));

end proc;

proc (A, B, L) local G; G := Groebner:-Basis(L, tdeg(x, y)); Groebner:-NormalForm(A+B, G, tdeg(x, y)) end proc

(36)

(37)

(38)

(39)

(40)

Multiplication

Procedure II.7

>	mum:=proc(A,B,L) local G;

>	G:=Basis(L,tdeg(x,y)); NormalForm(A*B,G,tdeg(x,y))

end proc;

proc (A, B, L) local G; G := Groebner:-Basis(L, tdeg(x, y)); Groebner:-NormalForm(A*B, G, tdeg(x, y)) end proc

(41)

a[1]*b[1]+(a[1]*b[2]+a[2]*b[1])*x+(a[1]*b[3]+a[3]*b[1])*y+(a[1]*b[6]+a[2]*b[3]+a[3]*b[2]+a[6]*b[1])*x*y+(a[3]*b[4]+a[1]*b[5]+a[5]*b[1]+a[4]*b[3]-a[2]*b[6]-a[6]*b[2])*y^3+(-a[2]*b[2]+a[4]*b[1]+a[3]*b[3]+a[1]*b[4])*y^2