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The degree of a monomial is defined as the sum of all the exponents of the variables, including the implicit exponents of 1 for the variables which appear without exponent; e.g., in the example of the previous section, the degree is + +. The degree of is 1+1+2=4. The degree of a nonzero constant is 0.
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer .
When a monomial order has been chosen, the leading monomial is the largest u in S, the leading coefficient is the corresponding c u, and the leading term is the corresponding c u u. Head monomial/coefficient/term is sometimes used as a synonym of "leading". Some authors use "monomial" instead of "term" and "power product" instead of "monomial".
For example (,,) = + is a multilinear polynomial of degree (because of the monomial ) whereas (,,) = + is not. The degree of a multilinear polynomial is the maximum number of distinct variables occurring in any monomial.
For example: is a term. The coefficient is −5, the indeterminates are x and y, the degree of x is two, while the degree of y is one. The degree of the entire term is the sum of the degrees of each indeterminate in it, so in this example the degree is 2 + 1 = 3. Forming a sum of several terms produces a polynomial.
To find the interpolation polynomial p(x) in the vector space P(n) of polynomials of degree n, we may use the usual monomial basis for P(n) and invert the Vandermonde matrix by Gaussian elimination, giving a computational cost of O(n 3) operations.
For any subset S of the variables appearing with nonzero exponent in the monomial, there is a contribution involving the product X S of those variables as term from e s (X 1, ..., X n), where s = #S, and the monomial X α / X S from h m − s (X 1, ..., X n); this contribution has coefficient (−1) s. The relation then follows from the ...
Both degree and dimension depend only on the set of the leading monomials of the Gröbner basis of the ideal for any monomial ordering. The dimension is the maximal size of a subset S of the variables such that there is no leading monomial depending only on the variables in S .