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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 .
The first (greatest) term of a polynomial p for this ordering and the corresponding monomial and coefficient are respectively called the leading term, leading monomial and leading coefficient and denoted, in this article, lt(p), lm(p) and lc(p). Most polynomial operations related to Gröbner bases involve the leading terms.
The leading entry (sometimes leading coefficient [citation needed]) of a row in a matrix is the first nonzero entry in that row. So, for example, in the matrix ( 1 2 0 6 0 2 9 4 0 0 0 4 0 0 0 0 ) , {\displaystyle {\begin{pmatrix}1&2&0&6\\0&2&9&4\\0&0&0&4\\0&0&0&0\end{pmatrix}},} the leading coefficient of the first row is 1; that of the second ...
Divide the previously dropped/summed number by the leading coefficient of the divisor and place it on the row below (this doesn't need to be done if the leading coefficient is 1). In this case q 3 = a 7 b 4 {\displaystyle q_{3}={\dfrac {a_{7}}{b_{4}}}} , where the index 3 = 7 − 4 {\displaystyle 3=7-4} has been chosen by subtracting the index ...
So if two leading coefficients are in the same column, then a row operation of type 3 could be used to make one of those coefficients zero. Then by using the row swapping operation, one can always order the rows so that for every non-zero row, the leading coefficient is to the right of the leading coefficient of the row above.
Let () be a polynomial equation, where P is a univariate polynomial of degree n.If one divides all coefficients of P by its leading coefficient, one obtains a new polynomial equation that has the same solutions and consists to equate to zero a monic polynomial.
Note: "lc" stands for the leading coefficient, the coefficient of the highest degree of the variable. This algorithm computes not only the greatest common divisor (the last non zero r i), but also all the subresultant polynomials: The remainder r i is the (deg(r i−1) − 1)-th subresultant polynomial.
A simplified version of the LLL factorization algorithm is as follows: calculate a complex (or p-adic) root α of the polynomial () to high precision, then use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm to find an approximate linear relation between 1, α, α 2, α 3, . . . with integer coefficients, which might be an ...