Multiplicity (mathematics)

In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion of multiplicity is important to be able to count correctly without specifying exceptions (for example, double roots counted twice). Hence the expression, "counted with multiplicity". If multiplicity is ignored, this may be emphasized by counting the number of distinct elements, as in "the number of distinct roots". However, whenever a set (as opposed to multiset) is formed, multiplicity is automatically ignored, without requiring use of the term "distinct". p-adic valuation In prime factorization, the multiplicity of a prime factor is its -adic valuation. For example, the prime factorization of the integer 60 is 60 = 2 × 2 × 3 × 5, the multiplicity of the prime factor 2 is 2, while the multiplicity of each of the prime factors 3 and 5 is 1. Thus, 60 has four prime factors allowing for multiplicities, but only three distinct prime factors. Let be a field and be a polynomial in one variable with coefficients in . An element is a root of multiplicity of if there is a polynomial such that and . If , then a is called a simple root. If , then is called a multiple root. For instance, the polynomial has 1 and −4 as roots, and can be written as . This means that 1 is a root of multiplicity 2, and −4 is a simple root (of multiplicity 1). The multiplicity of a root is the number of occurrences of this root in the complete factorization of the polynomial, by means of the fundamental theorem of algebra. If is a root of multiplicity of a polynomial, then it is a root of multiplicity of the derivative of that polynomial, unless the characteristic of the underlying field is a divisor of k, in which case is a root of multiplicity at least of the derivative. The discriminant of a polynomial is zero if and only if the polynomial has a multiple root. The graph of a polynomial function f touches the x-axis at the real roots of the polynomial.