SINGULAR                                 /  Development
 A Computer Algebra System for Polynomial Computations       /   version 4.1.1
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 by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann     \   Feb 2018
FB Mathematik der Universitaet, D-67653 Kaiserslautern        \

//First we define a ring in 3 variables; the monomial ordering in this case is "dp"
// You always need a ring before you start any computation with polynomials

ring r=0,(x,y,z),dp;

// Next we define a polynomial f and print it

poly f=x^3+y^6;

print(f);
y6+x3

// We define another polynomial g and perform a few elementary operations

poly g=z^7+11*x^4*y;

f+g;
z7+y6+11x4y+x3

f*g;
y6z7+11x4y7+x3z7+11x7y

//You can create lists of elements

list L = f,g,2*f+g;
L;
[1]:
   y6+x3
[2]:
   z7+11x4y
[3]:
   z7+2y6+11x4y+2x3
// And lists of lists:

list LL = L,L;
LL;
[1]:
   [1]:
      y6+x3
   [2]:
      z7+11x4y
   [3]:
      z7+2y6+11x4y+2x3
[2]:
   [1]:
      y6+x3
   [2]:
      z7+11x4y
   [3]:
      z7+2y6+11x4y+2x3

//From a mathematical point of view we can also define ideals and perform elementary operations
//with them
ideal I=f,g;
ideal J=f*f, f+g;

I;
I[1]=y6+x3
I[2]=z7+11x4y

> J;
J[1]=y12+2x3y6+x6
J[2]=z7+y6+11x4y+x3

I*J;
_[1]=y18+3x3y12+3x6y6+x9
_[2]=y6z7+y12+11x4y7+x3z7+2x3y6+11x7y+x6
_[3]=y12z7+11x4y13+2x3y6z7+22x7y7+x6z7+11x10y
_[4]=z14+y6z7+22x4yz7+11x4y7+121x8y2+x3z7+11x7y

I+J;
_[1]=y6+x3
_[2]=z7+11x4y
_[3]=y12+2x3y6+x6
_[4]=z7+y6+11x4y+x3

//Finally we can load libraries and call functions defined in them

LIB "CompAlg.LIB";

idealsum(I,3);
_[1]=y6+x3
_[2]=z7+11x4y

example idealsum;
// proc idealsum from lib Schreibtisch/CompAlgSingular/CompAlg.lib
EXAMPLE:
    ring r=0,(x,y),dp;
    poly f=x^2+y^3*x+y^5;
    poly g=y^7+x^5;
    ideal I=f,g,f*g,f^2;
    ideal J=idealsum(I,5);
    J;
J[1]=y5+xy3+x2
J[2]=y7+x5
J[3]=y12+xy10+x5y5+x6y3+x2y7+x7
J[4]=y10+2xy8+x2y6+2x2y5+2x3y3+x4