Leading Monomial

The leading monomial of a nonzero polynomial is its largest monomial with respect to a specified monomial order, with the coefficient omitted. For over a field , it is commonly denoted . The corresponding leading term includes its coefficient, so that , where is the leading coefficient.

The choice of monomial order is essential. For example, for over , the lexicographic order with gives , whereas the lexicographic order with gives . A nonzero constant polynomial has leading monomial 1. The leading monomial of the zero polynomial is left undefined.

For nonzero polynomials over a field, the compatibility of a monomial order with multiplication gives

Leading monomials determine which cancellations are possible in polynomial reduction and are central to Gröbner bases.