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For every monomial ordering, the empty set of polynomials is the unique Gröbner basis of the zero ideal. For every monomial ordering, a set of polynomials that contains a nonzero constant is a Gröbner basis of the unit ideal (the whole polynomial ring). Conversely, every Gröbner basis of the unit ideal contains a nonzero constant.
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term.Two definitions of a monomial may be encountered: A monomial, also called a power product or primitive monomial, [1] is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. [2]
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".
In mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists of all monomials.The monomials form a basis because every polynomial may be uniquely written as a finite linear combination of monomials (this is an immediate consequence of the definition of a polynomial).
It is a polynomial in which no variable occurs to a power of or higher; that is, each monomial is a constant times a product of distinct variables. For example f ( x , y , z ) = 3 x y + 2.5 y − 7 z {\displaystyle f(x,y,z)=3xy+2.5y-7z} is a multilinear polynomial of degree 2 {\displaystyle 2} (because of the monomial 3 x y {\displaystyle 3xy ...
Finally let B be a Gröbner basis of I for a monomial ordering refining the total degree partial ordering and G the (homogeneous) ideal generated by the leading monomials of the elements of B. The computation of the Hilbert series is based on the fact that the filtered algebra R/I and the graded algebras R/H and R/G have the same Hilbert series.