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Every Laurent polynomial can be written as a rational function while the converse is not necessarily true, i.e., the ring of Laurent polynomials is a subring of the rational functions. The rational function () = is equal to 1 for all x except 0, where there is a removable singularity. The sum, product, or quotient (excepting division by the ...
A polynomial function is one that has the form = + + + + + where n is a non-negative integer that defines the degree of the polynomial. A polynomial with a degree of 0 is simply a constant function; with a degree of 1 is a line; with a degree of 2 is a quadratic; with a degree of 3 is a cubic, and so on.
When the minimal polynomial is identical to the characteristic polynomial (the case k = 1), the Frobenius normal form is the companion matrix of the characteristic polynomial. As the rational canonical form is uniquely determined by the unique invariant factors associated to A, and these invariant factors are independent of basis, it follows ...
The word polynomial joins two diverse roots: the Greek poly, meaning "many", and the Latin nomen, or "name". It was derived from the term binomial by replacing the Latin root bi-with the Greek poly-. That is, it means a sum of many terms (many monomials). The word polynomial was first used in the 17th century. [6]
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.
Therefore, in computer algebra, normal form is a weaker notion: A normal form is a representation such that zero is uniquely represented. This allows testing for equality by putting the difference of two objects in normal form. Canonical form can also mean a differential form that is defined in a natural (canonical) way.
Plot of the absolute value of the seventh-order (n = 7) Chebyshev rational function for 0.01 ≤ x ≤ 100. Note that there are n zeroes arranged symmetrically about x = 1 and if x 0 is a zero, then 1 / x 0 is a zero as well. The maximum value between the zeros is unity. These properties hold for all orders. Defining:
The rational univariate representation or RUR is a representation of the solutions of a zero-dimensional polynomial system over the rational numbers which has been introduced by F. Rouillier. [10] A RUR of a zero-dimensional system consists in a linear combination x 0 of the variables, called separating variable, and a system of equations [11]