Laurent polynomial

In mathematics, a Laurent polynomial (namedafter Pierre Alphonse Laurent) in one variable over a field F {displaystyle mathbb {F} } is a linear combination of positive and negative powers of the variable with coefficients in F {displaystyle mathbb {F} } . Laurent polynomials in X form a ring denoted F {displaystyle mathbb {F} } . They differ from ordinary polynomials in that they may have terms of negative degree. The construction of Laurent polynomials may be iterated, leading to the ring of Laurent polynomials in several variables. Laurent polynomials are of particular importance in the study of complex variables. In mathematics, a Laurent polynomial (namedafter Pierre Alphonse Laurent) in one variable over a field F {displaystyle mathbb {F} } is a linear combination of positive and negative powers of the variable with coefficients in F {displaystyle mathbb {F} } . Laurent polynomials in X form a ring denoted F {displaystyle mathbb {F} } . They differ from ordinary polynomials in that they may have terms of negative degree. The construction of Laurent polynomials may be iterated, leading to the ring of Laurent polynomials in several variables. Laurent polynomials are of particular importance in the study of complex variables. A Laurent polynomial with coefficients in a field F {displaystyle mathbb {F} } is an expression of the form where X is a formal variable, the summation index k is an integer (not necessarily positive) and only finitely many coefficients pk are non-zero. Two Laurent polynomials are equal if their coefficients are equal. Such expressions can be added, multiplied, and brought back to the same form by reducing similar terms. Formulas for addition and multiplication are exactly the same as for the ordinary polynomials, with the only difference that both positive and negative powers of X can be present: