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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.
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.
By successively dividing out factors x − a, one sees that any polynomial with complex coefficients can be written as a constant (its leading coefficient) times a product of such polynomial factors of degree 1; as a consequence, the number of (complex) roots counted with their multiplicities is exactly equal to the degree of the polynomial.
Coefficient matrices are used in algorithms such as Gaussian elimination and Cramer's rule to find solutions to the system. The leading entry (sometimes leading coefficient [citation needed]) of a row in a matrix is the first nonzero entry in that row.
The theorem is used to find all rational roots of a polynomial, if any. It gives a finite number of possible fractions which can be checked to see if they are roots. If a rational root x = r is found, a linear polynomial ( x – r ) can be factored out of the polynomial using polynomial long division , resulting in a polynomial of lower degree ...
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.
For any given n ≥ 1, among the polynomials of degree n with leading coefficient 1 (monic polynomials): = is the one of which the maximal absolute value on the interval [−1, 1] is minimal. This maximal absolute value is: 1 2 n − 1 {\displaystyle {\frac {1}{2^{n-1}}}} and | f ( x ) | reaches this maximum exactly n + 1 times at: x = cos ...
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.