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MATH 819 Univariate Polynomials

创建日期：2024-08-01 15:10:52

所属分类：数学mathematics

标签： MATH 819

Univariate Polynomials

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MACM 401/MATH 701/MATH 819

Assignment

Due Monday February 4th at 4pm. Please hand in to Dropoff box 1b outside AQ 4100

Late Penalty: 20% for up to 48 hours late. Zero after that.

For problems involving Maple calculations and Maple programming, you should submit a

printout of a Maple worksheet of your Maple session.

Question 1 (15 marks): Univariate Polynomials

Reference section 2.5.

(a) Program the extended Euclidean algorithm for Q[x] in Maple. The input is two nonzero

polynomials a, b ∈ Q[x]. The output is three polynomials (s, t, g) where g is the

monic gcd of a and b and sa + tb = g holds.

Please print out the values of (rk, sk, tk) that are computed at each division step so

that we can observe the exponential growth in the size of the rational coefficients the

rk, sk, tk polynomials.

Use the Maple commands quo(a,b,x) and/or rem(a,b,x) to compute the quotient

and remainder of a divided b in Q[x]. Remember, in Maple, you must explicitly expand

products of polynomials using the expand(...) command.

Execute your Maple code on the following inputs.

> a := expand((x+1)*(2*x^4-3*x^3+5*x^2+3*x-1));

> b := expand((x+1)*(7*x^4+5*x^3-2*x^2-x+4));

Check that your output satisfies sa + tb = g and check that your result agrees with

Maple’s g := gcdex(a,b,x,’s’,’t’); command.

(b) Consider a(x) = x3 1, b(x) = x2 + 1, and c(x) = x2. Apply the algorithm in the proof

of Theorem 2.6 (as presented in class) to solve the polynomial diophantine equation

σa + τ b = c for σ, τ ∈ Q[x] satisfying deg σ < deg b deg g where g is the monic gcd

of a and b. Use Maple’s g := gcdex(a,b,x,’s’,’t’); command to solve sa + tb = g

for s, t ∈ Q[x] or your algorithm from part (a) above.

Question 2 (15 marks): Multivariate Polynomial Division

(a) Consider the polynomials

A = 6 y2x

2 + 5 yx2 + 3 xy2 + yx + y

2 + x + y and B = 2 yx2 + x + y.

Write A ∈ Z[y][x] and test if B|A by doing the division in Z[y][x] by hand. Show your

working. If B|A determine the quotient Q of A ÷ B. Check your answer using Maple’s

divide command.

(b) Given two polynomials A, B ∈ Z[x1, x2, . . . , xn] with B 6= 0, give pseudo code for the

multivariate division algorithm for dividing A by B. The pseudo code should begin

like this

Algorithm DIVIDE(A,B)

Inputs A, B ∈ Z[x1, x2, . . . , xn] satisfying B 6= 0 and n ≥ 0.

Output Q ∈ Z[x1, x2, . . . , xn] s.t. A = BQ or FAIL meaning B does not divide A.

I suggest you start with my pseudo code for the division algorithm in F[x] and modify

it to work in D[x1] where D = Z[x2, . . . , xn]. The algorithm will make a recursive call

to divide the leading coefficients in D. Because the algorithm is recursive you need a

base of the recursion.

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