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6.3 operations with polynomials and rational expressions


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6.3.1 var

var(rat)

:: Main variable (indeterminate) of rat.

return

indeterminate

rat

rational expression

[0] var(x^2+y^2+a^2);
x
[1] var(a*b*c*d*e);
a
[2] var(3/abc+2*xy/efg);
abc
References

ord, vars.


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6.3.2 vars

vars(obj)

:: A list of variables (indeterminates) in an expression obj.

return

list

obj

arbitrary

[0] vars(x^2+y^2+a^2);
[x,y,a]
[1] vars(3/abc+2*xy/efg);
[abc,xy,efg]
[2] vars([x,y,z]);
[x,y,z]
References

var, uc, ord.


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6.3.3 uc

uc()

:: Create a new indeterminate for an undermined coeficient.

return

indeterminate with its vtype 1.

[0] A=uc();
_0
[1] B=uc();
_1
[2] (uc()+uc())^2;
_2^2+2*_3*_2+_3^2
[3] (A+B)^2;
_0^2+2*_1*_0+_1^2
References

vtype, rtostr, strtov.


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6.3.4 coef

coef(poly,deg[,var])

:: The coefficient of a polynomial poly at degree deg with respect to the variable var (main variable if unspecified).

return

polynomial

poly

polynomial

var

indeterminate

deg

non-negative integer

[0] A = (x+y+z)^3;
x^3+(3*y+3*z)*x^2+(3*y^2+6*z*y+3*z^2)*x+y^3+3*z*y^2+3*z^2*y+z^3
[1] coef(A,1,y);
3*x^2+6*z*x+3*z^2
[2] coef(A,0);
y^3+3*z*y^2+3*z^2*y+z^3
References

var, deg, mindeg.


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6.3.5 deg, mindeg

deg(poly,var)

:: The degree of a polynomial poly with respect to variable.

mindeg(poly,var)

:: The least exponent of the terms with non-zero coefficients in a polynomial poly with respect to the variable var. In this manual, this quantity is sometimes referred to the minimum degree of a polynomial for short.

return

non-negative integer

poly

polynomial

var

indeterminate

[0] deg((x+y+z)^10,x);
10
[1] deg((x+y+z)^10,w);
0
[75] mindeg(x^2+3*x*y,x);
1

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6.3.6 nmono

nmono(rat)

:: Number of monomials in rational expression rat.

return

non-negative integer

rat

rational expression

[0] nmono((x+y)^10);
11
[1] nmono((x+y)^10/(x+z)^10);
22
[2] nmono(sin((x+y)^10));
1
References

vtype.


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6.3.7 ord

ord([varlist])

:: It sets the ordering of indeterminates (variables).

return

list of indeterminates

varlist

list of indeterminates

[0] ord();
[x,y,z,u,v,w,p,q,r,s,t,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,_x,_y,_z,_u,_v,
_w,_p,_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,_o,
exp(_x),(_x)^(_y),log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),
(-_x^2+1)^(-1/2),cosh(_x),sinh(_x),tanh(_x),
(_x^2+1)^(-1/2),(_x^2-1)^(-1/2)]
[1] ord([dx,dy,dz,a,b,c]);
[dx,dy,dz,a,b,c,x,y,z,u,v,w,p,q,r,s,t,d,e,f,g,h,i,j,k,l,m,n,o,_x,_y,
_z,_u,_v,_w,_p,_q,_r,_s,_t,_a,_b,_c,_d,_e,_f,_g,_h,_i,_j,_k,_l,_m,_n,
_o,exp(_x),(_x)^(_y),log(_x),(_x)^(_y-1),cos(_x),sin(_x),tan(_x),
(-_x^2+1)^(-1/2),cosh(_x),sinh(_x),tanh(_x),
(_x^2+1)^(-1/2),(_x^2-1)^(-1/2)]

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6.3.8 sdiv, sdivm, srem, sremm, sqr, sqrm

sdiv(poly1,poly2[,v])
sdivm(poly1,poly2,mod[,v])

:: Quotient of poly1 divided by poly2 provided that the division can be performed within polynomial arithmetic over the rationals.

srem(poly1,poly2[,v])
sremm(poly1,poly2,mod[,v])

:: Remainder of poly1 divided by poly2 provided that the division can be performed within polynomial arithmetic over the rationals.

sqr(poly1,poly2[,v])
sqrm(poly1,poly2,mod[,v])

:: Quotient and remainder of poly1 divided by poly2 provided that the division can be performed within polynomial arithmetic over the rationals.

return

sdiv(), sdivm(), srem(), sremm() : polynomial sqr(), sqrm() : a list [quotient,remainder]

poly1 poly2

polynomial

v

indeterminate

mod

prime

[0] sdiv((x+y+z)^3,x^2+y+a);    
x+3*y+3*z
[1] srem((x+y+z)^2,x^2+y+a);
(2*y+2*z)*x+y^2+(2*z-1)*y+z^2-a
[2] X=(x+y+z)*(x-y-z)^2;
x^3+(-y-z)*x^2+(-y^2-2*z*y-z^2)*x+y^3+3*z*y^2+3*z^2*y+z^3
[3] Y=(x+y+z)^2*(x-y-z);  
x^3+(y+z)*x^2+(-y^2-2*z*y-z^2)*x-y^3-3*z*y^2-3*z^2*y-z^3
[4] G=gcd(X,Y);
x^2-y^2-2*z*y-z^2
[5] sqr(X,G);
[x-y-z,0]
[6] sqr(Y,G);
[x+y+z,0]
[7] sdiv(y*x^3+x+1,y*x+1);  
divsp: cannot happen
return to toplevel
References

idiv, irem, %.


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6.3.9 tdiv

tdiv(poly1,poly2)

:: Tests whether poly2 divides poly1.

return

Quotient if poly2 divides poly1, 0 otherwise.

poly1 poly2

polynomial

[11] Y=(x+y+z)^5*(x-y-z)^3;  
x^8+(2*y+2*z)*x^7+(-2*y^2-4*z*y-2*z^2)*x^6
+(-6*y^3-18*z*y^2-18*z^2*y-6*z^3)*x^5
+(6*y^5+30*z*y^4+60*z^2*y^3+60*z^3*y^2+30*z^4*y+6*z^5)*x^3
+(2*y^6+12*z*y^5+30*z^2*y^4+40*z^3*y^3+30*z^4*y^2+12*z^5*y+2*z^6)*x^2
+(-2*y^7-14*z*y^6-42*z^2*y^5-70*z^3*y^4-70*z^4*y^3-42*z^5*y^2
-14*z^6*y-2*z^7)*x-y^8-8*z*y^7-28*z^2*y^6-56*z^3*y^5-70*z^4*y^4
-56*z^5*y^3-28*z^6*y^2-8*z^7*y-z^8
[12] for(I=0,F=x+y+z,T=Y; T=tdiv(T,F); I++); 
[13] I;
5
References

sdiv, sdivm, srem, sremm, sqr, sqrm.


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6.3.10 %

poly % m

:: integer remainder to all integer coefficients of the polynomial.

return

integer or polynomial

poly

integer or polynomial with integer coefficients

m

intger

[0] (x+2)^5 % 3;
x^5+x^4+x^3+2*x^2+2*x+2
[1] (x-2)^5 % 3;
x^5+2*x^4+x^3+x^2+2*x+1
[2] (-5) % 4;
3
[3] irem(-5,4);
-1
References

idiv, irem.


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6.3.11 subst, psubst

subst(rat[,varn,ratn]*)
psubst(rat[,var,rat]*)

:: Substitute ratn for varn in expression rat. (n=1,2,…. Substitution will be done successively from left to right if arguments are repeated.)

return

rational expression

rat ratn

rational expression

varn

indeterminate

[0] subst(x^3-3*y*x^2+3*y^2*x-y^3,y,2);
x^3-6*x^2+12*x-8
[1] subst(@@,x,-1);
-27
[2] subst(x^3-3*y*x^2+3*y^2*x-y^3,y,2,x,-1);
-27
[3] subst(x*y^3,x,y,y,x);  
x^4
[4] subst(x*y^3,y,x,x,y);    
y^4
[5] subst(x*y^3,x,t,y,x,t,y);
y*x^3
[6] subst(x*sin(x),x,t);
sint(t)*t
[7] psubst(x*sin(x),x,t);
sin(x)*t

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6.3.12 diff

diff(rat[,varn]*)
diff(rat,varlist)

:: Differentiate rat successively by var’s for the first form, or by variables in varlist for the second form.

return

expression

rat

rational expression which contains elementary functions.

varn

indeterminate

varlist

list of indeterminates

[0] diff((x+2*y)^2,x);  
2*x+4*y
[1] diff((x+2*y)^2,x,y);
4
[2] diff(x/sin(log(x)+1),x);
(sin(log(x)+1)-cos(log(x)+1))/(sin(log(x)+1)^2)
[3] diff(sin(x),[x,x,x,x]);
sin(x)

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6.3.13 ediff

ediff(poly[,varn]*)
ediff(poly,varlist)

:: Differentiate poly successively by Euler operators of var’s for the first form, or by Euler operators of variables in varlist for the second form.

return

polynomial

poly

polynomial

varn

indeterminate

varlist

list of indeterminates

[0] ediff((x+2*y)^2,x);  
2*x^2+4*y*x
[1] ediff((x+2*y)^2,x,y);
4*y*x

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6.3.14 res

res(var,poly1,poly2[,mod])

:: Resultant of poly1 and poly2 with respect to var.

return

polynomial

var

indeterminate

poly1 poly2

polynomial

mod

prime

[0] res(t,(t^3+1)*x+1,(t^3+1)*y+t);
-x^3-x^2-y^3

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6.3.15 fctr, sqfr

fctr(poly)

:: Factorize polynomial poly over the rationals.

sqfr(poly)

:: Gets a square-free factorization of polynomial poly.

return

list

poly

polynomial with rational coefficients

[0] fctr(x^10-1);
[[1,1],[x-1,1],[x+1,1],[x^4+x^3+x^2+x+1,1],[x^4-x^3+x^2-x+1,1]]
[1] fctr(x^3+y^3+(z/3)^3-x*y*z);
[[1/27,1],[9*x^2+(-9*y-3*z)*x+9*y^2-3*z*y+z^2,1],[3*x+3*y+z,1]]
[2] A=(a+b+c+d)^2;
a^2+(2*b+2*c+2*d)*a+b^2+(2*c+2*d)*b+c^2+2*d*c+d^2
[3] fctr(A);
[[1,1],[a+b+c+d,2]]
[4] A=(x+1)*(x^2-y^2)^2; 
x^5+x^4-2*y^2*x^3-2*y^2*x^2+y^4*x+y^4
[5] sqfr(A);
[[1,1],[x+1,1],[-x^2+y^2,2]]
[6] fctr(A);
[[1,1],[x+1,1],[-x-y,2],[x-y,2]]
References

ufctrhint.


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6.3.16 ufctrhint

ufctrhint(poly,hint)

:: Factorizes uni-variate polynomial poly over the rational number field when the degrees of its factors are known to be some integer multiples of hint.

return

list

poly

uni-variate polynomial with rational coefficients

hint

non-negative integer

[10] A=t^9-15*t^6-87*t^3-125;               
t^9-15*t^6-87*t^3-125
0msec
[11] N=res(t,subst(A,t,x-2*t),A);           
-x^81+1215*x^78-567405*x^75+139519665*x^72-19360343142*x^69
+1720634125410*x^66-88249977024390*x^63-4856095669551930*x^60
+1999385245240571421*x^57-15579689952590251515*x^54
+15956967531741971462865*x^51
...
+140395588720353973535526123612661444550659875*x^6
+10122324287343155430042768923500799484375*x^3
+139262743444407310133459021182733314453125
980msec + gc : 250msec
[12] sqfr(N);
[[-1,1],[x^81-1215*x^78+567405*x^75-139519665*x^72+19360343142*x^69
-1720634125410*x^66+88249977024390*x^63+4856095669551930*x^60
-1999385245240571421*x^57+15579689952590251515*x^54
...
-10122324287343155430042768923500799484375*x^3
-139262743444407310133459021182733314453125,1]]
20msec
[13] fctr(N);                               
[[-1,1],[x^9-405*x^6-63423*x^3-2460375,1],
[x^18-486*x^15+98739*x^12-9316620*x^9+945468531*x^6-12368049246*x^3
+296607516309,1],[x^18-8667*x^12+19842651*x^6+19683,1],
[x^18-324*x^15+44469*x^12-1180980*x^9+427455711*x^6+2793253896*x^3
+31524548679,1],
[x^18+10773*x^12+2784051*x^6+307546875,1]]
167.050sec + gc : 1.890sec
[14] ufctrhint(N,9);
[[-1,1],[x^9-405*x^6-63423*x^3-2460375,1],
[x^18-486*x^15+98739*x^12-9316620*x^9+945468531*x^6-12368049246*x^3
+296607516309,1],[x^18-8667*x^12+19842651*x^6+19683,1],
[x^18-324*x^15+44469*x^12-1180980*x^9+427455711*x^6+2793253896*x^3
+31524548679,1],
[x^18+10773*x^12+2784051*x^6+307546875,1]]
119.340sec + gc : 1.300sec
References

fctr, sqfr.


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6.3.17 modfctr

modfctr(poly,mod)

:: Factorizer over small finite fields

return

list

poly

Polynomial with integer coefficients

mod

non-negative integer

[0] modfctr(x^10+x^2+1,2147483647);
[[1,1],[x+1513477736,1],[x+2055628767,1],[x+91854880,1],
[x+634005911,1],[x+1513477735,1],[x+634005912,1],
[x^4+1759639395*x^2+2045307031,1]]
[1] modfctr(2*x^6+(y^2+z*y)*x^4+2*z*y^3*x^2+(2*z^2*y^2+z^3*y)*x+z^4,3);
[[2,1],[2*x^3+z*y*x+z^2,1],[2*x^3+y^2*x+2*z^2,1]]
References

fctr, sqfr.


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6.3.18 ptozp

ptozp(poly)

:: Converts a polynomial poly with rational coefficients into an integral polynomial such that GCD of all its coefficients is 1.

return

polynomial

poly

polynomial

[0] ptozp(2*x+5/3);
6*x+5
[1] nm(2*x+5/3);   
2*x+5/3
References

nm, dn.


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6.3.19 prim, cont

prim(poly[,v])

:: Primitive part of poly.

cont(poly[,v])

:: Content of poly.

return poly

polynomial over the rationals

v

indeterminate

[0] E=(y-z)*(x+y)*(x-z)*(2*x-y);
(2*y-2*z)*x^3+(y^2-3*z*y+2*z^2)*x^2+(-y^3+z^2*y)*x+z*y^3-z^2*y^2
[1] prim(E);
2*x^3+(y-2*z)*x^2+(-y^2-z*y)*x+z*y^2
[2] cont(E);
y-z
[3] prim(E,z);
(y-z)*x-z*y+z^2
References

var, ord.


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6.3.20 gcd, gcdz

gcd(poly1,poly2[,mod])
gcdz(poly1,poly2)

:: The polynomial greatest common divisor of poly1 and poly2.

return

polynomial

poly1 poly2

polynomial

mod

prime

[0] gcd(12*(x^2+2*x+1)^2,18*(x^2+(y+1)*x+y)^3);
x^3+3*x^2+3*x+1
[1] gcdz(12*(x^2+2*x+1)^2,18*(x^2+(y+1)*x+y)^3);
6*x^3+18*x^2+18*x+6
[2] gcd((x+y)*(x-y)^2,(x+y)^2*(x-y));
x^2-y^2
[3] gcd((x+y)*(x-y)^2,(x+y)^2*(x-y),2);
x^3+y*x^2+y^2*x+y^3
References

igcd,igcdcntl.


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6.3.21 red

red(rat)

:: Reduced form of rat by canceling common divisors.

return

rational expression

rat

rational expression

[0] (x^3-1)/(x-1);
(x^3-1)/(x-1)
[1] red((x^3-1)/(x-1));
x^2+x+1
[2] red((x^3+y^3+z^3-3*x*y*z)/(x+y+z));
x^2+(-y-z)*x+y^2-z*y+z^2
[3] red((3*x*y)/(12*x^2+21*y^3*x));
(y)/(4*x+7*y^3)
[4] red((3/4*x^2+5/6*x)/(2*y*x+4/3*x));
(9/8*x+5/4)/(3*y+2)
References

nm, dn, gcd, gcdz, ptozp.


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