HP 48gII hp 48gII_user's manual_English_E_HDPMSG48E67_V2.pdf - Page 189

Note: Refer to the help facility in the calculator for description and examples on other modular arithmetic. Many of these functions are applicable to polynomials. For information on modular arithmetic with polynomials please refer to a textbook on number theory. Polynomials Polynomials are algebraic expressions consisting of one or more terms containing decreasing powers of a given variable. For example, 'X^3+2*X^2-3*X+2' is a third-order polynomial in X, while 'SIN(X)^2-2' is a second-order polynomial in SIN(X). A listing of polynomial-related functions in the ARITHMETIC menu was presented earlier. Some general definitions on polynomials are provided next. In these definitions A(X), B(X), C(X), P(X), Q(X), U(X), V(X), etc., are polynomials. • Polynomial fraction: a fraction whose numerator and denominator are polynomials, say, C(X) = A(X)/B(X) • Roots, or zeros, of a polynomial: values of X for which P(X) = 0 • Poles of a fraction: roots of the denominator • Multiplicity of roots or poles: the number of times a root shows up, e.g., P(X) = (X+1)2(X-3) has roots {-1, 3} with multiplicities {2,1} • Cyclotomic polynomial (Pn(X)): a polynomial of order EULER(n) whose roots are the primitive n-th roots of unity, e.g., P2(X) = X+1, P4(X) = X2+1 • Bézout's polynomial equation: A(X) U(X) + B(X)V(X) = C(X) Specific examples of polynomial applications are provided next. Modular arithmetic with polynomials The same way that we defined a finite-arithmetic ring for numbers in a previous section, we can define a finite-arithmetic ring for polynomials with a given polynomial as modulo. For example, we can write a certain polynomial P(X) as P(X) = X (mod X2), or another polynomial Q(X) = X + 1 (mod X-2). A polynomial, P(X) belongs to a finite arithmetic ring of polynomial modulus M(X), if there exists a third polynomial Q(X), such that (P(X) - Q(X)) is a multiple of M(X). We then would write: P(X) ≡ Q(X) (mod M(X)). The later expression is interpreted as "P(X) is congruent to Q(X), modulo M(X)". Page 5-18