Computer Algebra for Computer Engineers

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1 p.1/14 Computer Algebra for Computer Engineers Preliminaries Priyank Kalla Department of Electrical and Computer Engineering University of Utah, Salt Lake City

2 p.2/14 Notation R: Real Numbers Q: Fractions C: Complex Z: Integers N: Natural numbers? Z + denotes positive integers {1, 2,...}, and non-negative integers are {0, 1,...}. Z n : Z mod n = {0, 1,...,n 1}

3 p.3/14 Group An Abelian group is a set G and a binary operation + satisfying all the following properties: Closure: For every a,b G,a + b G. Associativity: For every a,b,c G,a + (b + c) = (a + b) + c. Commutativity: For every a,b G,a + b = b + a. Identity: There is an identity element 0 G such that for all a G;a + 0 = a. Inverse: If a G, then there is an element a 1 G such that a + a 1 = 0. Examples of group operations: Go to Wikipedia and find out

4 p.4/14 Rings A Commutative ring with unity is a set R and two binary operations + and, as well as two distinguished elements 0, 1 R such that: R is an Abelian group with respect to addition with additive identity element 0. Multiplication Closure: For every a,b R, a b R. Multiplication Associativity: For every a, b, c R, a (b c) = (a b) c. Multiplication Commutativity: For every a,b R, a b = b a. Multiplication Identity: There is an identity element 1 R such that for all a R, a 1 = a. Distributivity: For every a,b,c R, a (b + c) = a b + a c holds for all a,b,c R.

5 p.5/14 Fields A field F is a commutative ring with unity, where every element in F except 0 has a multiplicative inverse, i.e. a F {0}, â F such that a â = 1. R,Q,C,Z p, where p = prime are fields Arbitrary Z n (i.e. when n p) is not a field Finite fields are also called Galois fields: GF(q) or F q where q = p m. Examples of GF: Z 2,Z 3,Z 5,...,Z p,f 4,F 8,F 9,F 16,...,F q where q = p m Note: Z 2 = F 2 ;Z 3 = F 3 ;...;Z p = F p ; but Z 4 F 4 Fields are unique (up to the labeling of elements).

6 p.6/14 Modulo Arithmetic The set Z n = {0, 1,...,n 1}, where n N, forms a commutative ring with unity. It is called the residue class ring, where addition and multiplication are defined modulo n (modn). Integers x,y are called congruent modulo n (x y mod n) if n is a divisor of their difference: n (x y). (a + b)%n = (a%n + b)%n = (a + b%n)%n = (a%n + b%n)%n (1) (a b)%n = (a%n b)%n = (a b%n)%n = (a%n b%n)%n (2) ( a)%n = (n a)%n (3)

7 p.7/14 Polynomial Rings k[x] denotes the ring of polynomials with coefficients from the field k. Customary to use k for infinite fields, and F q for finite fields Examples: R[x], Q[x], F q [x], Z n [x] Similarly, k[x 1,...,x n ] is the ring of multivariate polynomials with coefficients in k Note k[x 1,...,x n ] is itself a ring, not a field

8 p.8/14 Polynomials Let R be a ring. A polynomial over R in the indeterminate x is an expression of the form a 0 + a 1 x + a 2 x a n x n = a i x i, a i R. Elements a i are coefficients, n is the degree. The element a n is called the leading coefficient; when a n = 1, the polynomial is monic. We ll take a look at multi-variate polynomials a little later. Ordering the monomials of a multi-variate polynomial is very important. Polynomial manipulation is carried out according to the given order. Monomial order defines: Leading terms, leading coefficients, leading monomials, etc. This is straightforward in univariate polynomials. If f(x) = a n x n + + a 1 x + a 0, then LT(f) = a n x n, LC(f) = a n, and LM(f) = x n.

9 p.9/14 Affine Space Given a field k and n Z +, we define n-dimensional affine space over k to be the set: k n = {(a 1,...,a n ) : a 1,...,a n k} In case k 2 = R 2, we get our affine plane. Affine spaces relate to polynomials: A polynomial f = a i x i k[x 1,...,x n ] also gives a corresponding function f : k n k. Hence, the notion of a polynomial function!

10 p.10/14 Polynomials and Functions So when we say, is f = 0?, it may mean Is f the zero polynomial? i.e. are all coefficients zero? Or does f represent the zero function over k n k? For example: F = x 2 + x Z 2 [x]. Note that F is a symbolically non-zero polynomial, but it induces the zero function over f : Z 2 Z 2. Proposition 1 (CLO-Book) Let k be an infinite field, and f k[x 1,...,x n ]. Then f = 0 f : k n k is the zero function. Also, f = g f : k n k and g : k n k are equal as functions. Theorem 1 Fundamental Theorem of Algebra: Every non-constant polynomial f C[x] has a root in C. Note: Algebraically closed fields!

11 p.11/14 Variety Let k be a field, and f 1,...,f s are polynomials in k[x 1,...,x n ]. Then we set: V(f 1,...,f s ) = {(a 1,...,a n ) k n : f i (a 1,...,a n ) = 0,1 i s} We call V(f 1,...,f s ) the affine variety defined by the polynomials. Variety = Set of all solutions to a given set of polynomial equations! Look at some varieties...

12 p.12/14 Variety Some Notes In V (f 1,...,f s ), s = given (finite)! Variety = set of points (that are the solutions to a finite system of equations) Is (a 1,...,a n ) k n an affine variety? Let k = R and let A = {(x,y) R 2 : y > 0}. Not an affine variety. Why?

13 Union & Intersection of Varieties lemma 1 Let V,W k n are affine varieties, then V W and V W are also affine varieties; where: Let V = V(f 1,...,f s ) and W = V(g 1,...,g t ) V W = V(f 1,...,f s,g 1,...,g t ) V W = V(f i g j : 1 i s,1 j t) Q: Is every finite subset of k n an affine variety? Q: Is every finite union/intersection of affine varieties also an affine variety? Infinite union? Q: Given f 1,...,f s, do they have a common solution? Do they have finite solutions? bases. We can answer these questions using Gröbner p.13/14

14 p.14/14 Ideals Definition 1 A subset I R = k[x 1,...,x n ] is an ideal if: 0 I If f,g I, then f + g I If f I and h R then f h I Definition 2 Let f 1,f 2,...,f s k[x 1,...,x n ]. Let s f 1,f 2...,f s = { f i g i : h 1,...h s k[x 1,...,x n ]} (4) i=1 I = f 1,f 2...,f s is an ideal generated by f 1,...,f s and the polynomials are called the generators.

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