5_things_I'm_glad_I_learned_SGW2011

---

title: 5 things I'm glad I learned SGW2011 permalink: wiki/5_things_I'm_glad_I_learned_SGW2011/ layout: wiki

Return to the Main SGW2011 page

Please write (like an answer to a FAQ) 5 things you can't believe you did not know about M2.

Courtney Gibbons

Branden Stone shared a neat trick with me this time! Let's say you want a function to tell the user something about the input. You can do this in one line using C syntax (<<). For example, given the function f as below,

                           f = t -> (
                           if even t == true then   (<< "Your number " << t << " is even." << endl)
                           else (<< "Your number " << t << "i s odd." << endl)
                           )

evaluating f(6) and f(19) yields

                           i10 : f(6)
                           Your number 6 is even.
                           
                           i11 : f(19)
                           Your number 19 is odd.

I wish I had known about the command prune earlier in my M2 career. Consider a module generated as the cokernel of a random map as below:

                           R = ZZ/101[x,y];
                           S = R/ideal(x^2);
                           M = coker random(S^1,S^{5:-3})
                           prune M
                           betti res (M, LengthLimit => 5)
                           betti res (prune M, LengthLimit => 5)

The output shows that the cokernel of the random map isn't minimally presented by default:

                           i3 : M = coker random(S^1,S^{5:-3})
                           o3 = cokernel | -11xy2-5y3 -33xy2-19y3 32xy2-25y3 26xy2-y3 -32xy2+46y3 |
                                                       1
                           o3 : S-module, quotient of S
                           i4 : prune M
                           o4 = cokernel | y3 xy2 |
                                                       1
                           o4 : S-module, quotient of S

At an earlier workshop, I learned two very useful commands: apply and peek. With respect to apply, I may never write a for loop again. And with peek, I will always be able to figure out the keys of a hash table!

Branden Stone

• To determine if an element f is in an ideal I, use the colon I:f; if you get the unit ideal, you know that f is an element of I
• If an element f is in an ideal I, you can determine the coefficients of the generators of I needed to write f as a combination of the generators of I; below is an example:

                                           i1 : R = ZZ/101[x,y]
                                           
                                           o1 = R
                                           
                                           o1 : PolynomialRing
                                           
                                           i2 : I = ideal"x+y,xy-y2,x3"
                                           
                                                                     2   3
                                           o2 = ideal (x + y, x*y - y , x )
                                           
                                           o2 : Ideal of R

Now consider the element f below; if we colon it in to I, we get the unit ideal. This tells us that f is an element of I.

                                           i3 : f = x^3*y+x*y^3
                                           
                                                 3       3
                                           o3 = x y + x*y
                                           
                                           o3 : R
                                           
                                           i4 : I:f
                                           
                                           o4 = ideal 1

Using the // command, we can find the desired coefficients:

                                           o4 : Ideal of R
                                           
                                           i5 : f//gens I
                                           
                                           o5 = {1} | x2y-xy2+y3 |
                                                {2} | y2         |
                                                {3} | 0          |
                                           
                                                        3       1
                                           o5 : Matrix R  <--- R  

Thus, we can write our element as follows:

                       f = x^3*y + x*y^3 = (x^2*y - x*y^2 + y^3)*(x+y) + y^2*(x*y-y^2) + 0*x^3 
                    

• To get the elements from a matrix M, use: flatten entries M
• How to comment out a block of code: {* comments here *}
• Debugging: For example, in emacs, when an error occurs you can goto the error message and hit return to go to the error in your code.