polynomial3 http://www.openmath.org/cd/polynomial3.ocd 2006-07-12 2004-07-12 0 0 experimental alg1 arith1 logic1 quant1 set1 setname1 setname2 relation1 fns1 interval1 integer1 polynomial1 polynomial2 This CD holds a collection of basic modular arithmetic for polynomials over fields. The data structures for polynomials can be arithmetic expressions, for instance using the ring1.expression symbol, or DMP as in the CD polyd1. gcd The n-ary greatest common divisor for univariate polynomials over fields. The gcd(X,Y,Z). factors This symbol is a unary function, whose argument should be a polynomial f. When applied to f, it represents a complete list of irreducible factors of f. The following expression represents the list [X+1,X+1] of rational polynomials. 2 2 1 quotient application This symbol represents the binary division operator on univariate polynomials over fields. That is, for univariate polynomials a and b, quotient(a,b) denotes the polynomial q such that a=b*q+r, with degree(r) less than degree(b). For all a,b with a,b univariate polynomials over a field F we have a = b * quotient(a,b) + remainder(a,b) and degree(remainder(a,b)) is less than degree(b). 1 remainder application The symbol represents a binary function, whose arguments should be univariate polynomials in the same polynomial ring whose coefficient ring is a field. When applied to a and b, it represents the polynomial remainder after division of a by b. For univariate polynomials a and b, remainder(a,b) denotes r such that a=b*q+r, with degree(r) less than degree(b). See remainder for a formal statement of this property.