[1] Let R = k[x

- (a) Show that dim
_{k}R_{d}= dim_{k}S_{d}. - (b) Show that dim
_{k}S_{d}is equal to the binomial coefficient "n+d choose d". [Hint: count the number of monomials of degree d by writing each monomial as a product, not using exponents. For example, say n = 3, and consider x_{0}^{2}x_{2}x_{3}^{3}to be x_{0}x_{0}x_{2}x_{3}x_{3}x_{3}. But you can indicate the same information by replacing the variables by *'s, and separating the different variables by |'s. So x_{0}x_{0}x_{2}x_{3}x_{3}x_{3}becomes **||*|***, meaning two x_{0}'s, no x_{1}'s, one x_{2}, and three x_{3}'s.]

- (a) Find each singular point p of F = x^2*y - y^2 + 2*y - 1,
and for each one, find its multiplicity m
_{p}(F) and all tangent lines at p. Check your answer using`Macaulay2`. - (b) Use
`Macaulay2`to find the singular points of F = x^4 + 2*x^2*y^2 + 2*x^3 + 2*x*y^2 + x^2 + y^2. For each singular point p, find its multiplicity m_{p}(F) and all tangent lines at p. - Notes: Recall that a point p is
*singular*if and only if p is in V(F, F_{x}, F_{y}), where F_{x}and F_{y}represent partial derivatives with respect to x and y, resp. In`Macaulay2`, for any ideal I, we can find V(I) via the command`decompose I`. You can also use`Macaulay2`to simplify substitutions: given F(x,y) = x^2 - y^3, to find F(x+1,y-3), simply input (x+1)^2 - (y-3)^3, as in the following example:3 2 2 i11 : (x+1)^2 - (y-3)^3 o11 = - y + x + 9y + 2x - 27y + 28

- (a) F = y^2-x^5, G = y^2*x^2-x^5-y^5
- (b) F = y^2-x^3, G = (y^2+x^2)^2+3*x^2*y-y^3
- (c) F = y^4+x^4-x^3 G = y^3*x^2+y^5+x^2

Now,

findIntMult = (I,P,R) -> ( if(dim(R/mylocalize(I,P,R)) > 0 ) then print "The intersection multiplicity is infinite!!!!!!" else degree( R/mylocalize(I,P,R) )) mylocalize = (I,P,R) -> ( L:=primaryDecomposition I; j:=-1; J:= I; B:= ideal (matrix {{1}}**R); -- Make B the ideal (1) but in current ring, not ZZ. scan(#L, i->( J=(L_i)*ideal (matrix {{1}}**R); -- convert L_i into an ideal if it's of type MonomialIdeal if( (P:J) == (ideal (matrix {{1}}**R)) ) then B=intersect(B,J); )); B)Sample Session:

Macaulay 2, version 0.9.2 --Copyright 1993-2001, D. R. Grayson and M. E. Stillman --Singular-Factory 1.3c, copyright 1993-2001, G.-M. Greuel, et al. --Singular-Libfac 0.3.2, copyright 1996-2001, M. Messollen i1 : R = QQ[x,y] o1 = R o1 : PolynomialRing i2 : findIntMult = (I,P,R) -> ( if(dim(R/mylocalize(I,P,R)) > 0 ) then print "The intersection multiplicity is infinite!!!!!!" else degree( R/mylocalize(I,P,R) )) mylocalize = (I,P,R) -> ( L:=primaryDecomposition I; j:=-1; J:= I; B:= ideal (matrix {{1}}**R); -- Make B the ideal (1) but in current ring, not ZZ. scan(#L, i->( J=(L_i)*ideal (matrix {{1}}**R); -- convert L_i into an ideal if it's of type MonomialIdeal if( (P:J) == (ideal (matrix {{1}}**R)) ) then B=intersect(B,J); )); B) o2 = findIntMult o2 : Function i3 : o3 = mylocalize o3 : Function i4 : findIntMult(ideal(x^3,y^4),ideal(x,y),R) o4 = 12 i5 : quitNotes:

(1)

(2) The colon in

(3) Now,

(4) P:J is the usual colon operation in commutative algebra; it is the ideal of all f in R such that fJ is contained in P. Thus P:J = (1) if and only if P contains J, so