Test 5: Elimination and Nullstellensatz

How to answer ? Make the necessary computations.
Using comments is always better

(* this is a comment just put it inside (*  *) and Mathematica will not pay attention to this *)

How to submit ? When you have finished, CLEAN your file (delete unnecessary computations)
Save it and send it to my e-mail address: dahan@math.kyushu-u.ac.jp

You can answer the questions on a paper and submit it (with the other part of the test). But always make the big computations on Mathematica.

Good luck.

Exercise 1 (Vanishing polynomial of algebraic numbers)

Let a ((=)^3)√ 7=(7^(1/3))  and b=(e^(2iπ/5))=cos(2π/5)+i sin(2π/5). They are algebraic numbers. Write down their minimal polynomial Pa and Pb.

HELP: Pa has degree 3 and Pb has degree 4.

HELP 2 : The command substitute will be usefull : for example to compute if P(X) is a function, then to compute  P(X+a Y) just write P/.X->X+a Y
HELP 3: the degree of the polynomials are never larger than 24.

Pa=

Pb=

Let c=a+b and d=a b. Compute  vanishing polynomials Pc and Pd of c and d. Use the resultant if necessary (Cf. Lect VI,  Part II, Paragraph 2 and file "VanishPolyOnAlgNbr.nb" downloadable on Yokoyama's webpage).

Let f=2+2c+(c^2). Compute a vanishing polynomial Pf of f.

Exercise 2 (Radical membership)

Consider the system F hereunder:
Let I = ⟨F⟩ ⊂ ℂ[x, y] the ideal it generates over ℂ.
Let also f the polynomial hereunder.

F={f1=y^2-y,f2=-125 (x^3)+75 (x^4)-15 (x^5)+(x^6)+(x^2) y+123 (x^3) y-74 (x^4) y+15 (x^5) y-(x^6) y}

f=-5 x+(x^2)+4 x y

Show that f∉I (use ideal membership, Lect.  IV)

But show that f∈ √I  (the radical of I. Use Rabonovitch's trick, Lect VII Slide 18)

Exercise 3 (elimination of system 2-2 with resultant)

Consider the 2 polynomials A and  in ℚ[x,y]

A=Collect[-1-y+(x^2) y,y]

B=Collect[-(x^2)-y-(x^2) y+(x^6) y-(y^2)+(x^4) (y^2),y]

Plot the 2 curves (over the range x=-3,..,3 and y=3,..,3)

Compute the resultant r of A and B that eliminates y

Compute the roots of r.

Are there intersection points in (ℝ^2)?

Are there intersection points in (ℂ^2) ?

Exercise 4 (elimination, system 3-3)

We consider the following system of equations F = {(Subscript[f, 1]),(Subscript[f, 2]),... , (Subscript[f, 10])} of ℚ[x,y,z].
The variables are ordered such that x>y>z.

F={f1=2-z-2 (z^2)+(z^3),
f2=10+3 x-5 y+3 x z-3 y z-10 (z^2)+2 y (z^2),
f3=10-x-y+x z-y z-10 (z^2)+2 x (z^2),
f4=-4-3 x+(y^2)+2 z-3 x z+(y^2) z+6 (z^2),f5=8+x-4 y+x y+2 z+x z-4 y z+x y z-6 (z^2),f6=12-x+(x^2)-2 y+2 z-x z+(x^2) z-2 y z-10 (z^2),f7=-22-15 x-3 y+3 (y^2)+3 (y^3)+12 z-15 x z+3 y z+34 (z^2),f8=314+163 x-18 (x^2)-181 y+42 x y-102 (y^2)+30 x (y^2)+12 z+73 x z-31 y z-302 (z^2),
f9=-114+97 x-12 (x^2)+281 y-222 x y+30 (x^2) y+12 (y^2)-132 z+37 x z+41 y z-18 (z^2),
f10=358+1221 x-306 (x^2)+30 (x^3)-147 y-36 x y-264 (y^2)-516 z+441 x z+93 y z-874 (z^2)}

Let I=,⟨F⟩ the ideal generated. Compute a basis of the first and second elimination ideal of I (find 2 families of polynomials (Subscript[G, 1]) and (Subscript[G, 2]) such that   (Subscript[E, 1])(I)=⟨(Subscript[G, 1])⟩
and (Subscript[E, 2])(I)=⟨(Subscript[G, 2])⟩, where (Subscript[E, 1])(I)=I⋂ℚ[y,z]  and  (Subscript[E, 2])(I)=I⋂ℚ[z])

HELP: use the elimination theorem.

Check that z=1 ∈(Subscript[𝕍, ℂ])((Subscript[E, 2])(I)). Find all the solutions (a,b,c)∈(ℂ^3)of F,
such that c=1 (find the solutions that extends the partial solution 1).

HELP: substite the partial solution z=1 in the family (Subscript[G, 2]), and solve in y....then finally solve in x.