Apply sturms sequences and find out exactly how many

Question

Math and Computers, Math 165 homework 2-

1. Using Descartes' rule of signs find as much information as you can about the possible number of roots (counting multiplicities) of each of the following polynomials:

a) x⁴ - x² + x - 2

b) x⁹ - x⁵ + x² + 2

c) x⁵ + 2x³ - x² + x - 1

2. Apply Sturm's sequences and find out exactly how many distinct roots are there for each of the polynomials of problem one.

3. Is the polynomial x² - 4 in the ideal generated by the polynomials x³ + x² - 4x - 4, x³ - x² - 4x + 4, x³ - 2x² - x + 2?

4. Explain why GCD(f, g, h) = GCD(GCD(f, g), h). Also explain why for univariate polynomials the ideal (f₁, f₂, . . . , f_k) is equal to (GCD(f₁, f₂, . . . , f_k)).

5. Sketch the following affine varieties (or at least the real parts of it!). in R²:

a) V (x² - y²),

b) V (x² + 4y² + 2x - 16y + 1) in R³:

c) V (xz² - xy),

d) V (x⁴ - zx, x³ - yz).

6. Consider the set {(x, x): x ∈ R, x ≠ 1} ⊂ R². This is a straight line minimus a point. Show that this set is not an affine variety (Hint: Prove that if a polynomial vanishes at the set must also vanish at (1, 1).

7. The basis of an ideal is different from a basis in linear algebra in that we do not care about linear independence! As a consequence when we write an element f ∈ (f₁, . . . , f_s) as f = ∑h_if_i the coefficients h_i are not always unique. As an example, write x² + xy + y² ∈ (x, y) in two different ways.

8. Each of the following polynomials is written with its monomials ordered according to exactly one of the monomial orders: Lex, graded lex, or graded reverse lex. Determine which monomial order was used in each case.

(a) 7x²y⁴z-2xy⁶+x²y²

(b) xy³z+xy²z²+x²z³

(c) x⁴y⁵z+2x³y²z-4xy²z⁴

9. Show that graded reverse lexicographic order is indeed a monomial order.

10. Let > be a monomial order in S = C[x₁, . . . , x_n].

(a) Let f ∈ S and let m be a monomial. Show that LT(m·f) = m·LT(f).

(b) Let f, g ∈ S. Is LT(f · g) necessarily the same as LT(f) · LT(g)?