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Manipulating polynomials in generalized form

Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: Let be an integral domain, P the integral domain of polynomials over.

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manipulating polynomials in generalized form

Save to Library. Create Alert. Launch Research Feed. Share This Paper. Top 3 of Citations View All Automatic parameterization of rational curves and surfaces IV: algebraic space curves. Abhyankar, C. Bajaj TOGS Quantifier elimination for real closed fields by cylindrical algebraic decomposition.

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Some features of the site may not work correctly. DOI: We present a novel method to design and optimize window functions based on combinations of linearly independent functions. View on IEEE. Save to Library. Create Alert. Launch Research Feed. Share This Paper. Figures and Tables from this paper. Figures and Tables. References Publications referenced by this paper. New efficient window function, replacement for the hamming window. Research Feed.

Highly Influential. View 5 excerpts, references background and methods. Adjustable window based design of multiplier-less cosine modulated filter bank using swarm optimization algorithms. Polynomial windows with low sidelobes' level. View 2 excerpts, references background.

Adjustable high resolution window function. View 1 excerpt, references background. Variable window length applied to a modified Hanning filter for optimal amplitude estimation of power systems signals.Algebraic number theory.

Noncommutative algebraic geometry. In mathematicsespecially in the field of algebraa polynomial ring or polynomial algebra is a ring which is also a commutative algebra formed from the set of polynomials in one or more indeterminates traditionally also called variables with coefficients in another ringoften a field.

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Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers. Polynomial rings occur and are often fundamental in many parts of mathematics such as number theorycommutative algebra and ring theory and algebraic geometry.

Many classes of rings, such as unique factorization domainsregular ringsgroup ringsrings of formal power seriesOre polynomialsgraded ringshave been introduced for generalizing some properties of polynomial rings. A closely related notion is that of the ring of polynomial functions on a vector spaceand, more generally, ring of regular functions on an algebraic variety.

The polynomial ringK [ X ]in X over a field or, more generally, a commutative ring K can be defined [1] there are other equivalent definitions that are commonly used as the set of expressions, called polynomials in Xof the form. The symbol X is called an indeterminate [2] or variable. However, here, X has not any value other than itselfand cannot vary, being a constant in the polynomial ring.

Two polynomials are equal when the corresponding coefficients of each X k are equal. One can think of the ring K [ X ] as arising from K by adding one new element X that is external to Kcommutes with all elements of Kand has no other specific properties.

This may be used for defining polynomial rings. The polynomial ring in X over K is equipped with an addition, a multiplication and a scalar multiplication that make it a commutative algebra.

These operations are defined according to the ordinary rules for manipulating algebraic expressions. Specifically, if. In these formulas, the polynomials p and q are extended by adding "dummy terms" with zero coefficients, so that all p i and q i that appear in the formulas are defined. It is straightforward to verify that these three operations satisfy the axioms of a commutative algebra over K.

Therefore, polynomial rings are also called polynomial algebras. In this case, p 0 and X are considered as alternate notations for the sequences p 00, 0, A straightforward use of the operation rules shows that the expression. The constant term of p is p 0. The leading coefficient of p is p m. A nonzero polynomial is monic if its leading coefficient is 1. Given two polynomials p and qone has. It follows immediately that, if K is an integral domain, then so is K [ X ].Based on the Lagrange inversion theorem and the theorem of logarithmic derivative for generating functions, we obtain new properties related to the compositional inverse generating functions of those polynomials.

We apply those results for obtaining explicit formulas and identities for such polynomials as the generalized Bernoulli, generalized Euler, Frobenius-Euler, generalized Sylvester, generalized Laguerre, Abel, Bessel, Stirling, Narumi, Peters, Gegenbauer, and Meixner polynomials.

Polynomials - Theory and Application. Generating functions are a powerful tool for solving problems in number theory, combinatorics, algebra, probability theory, and other fields of mathematics. One of the advantages of generating functions is that an infinite number sequence can be represented in a form of a single expression.

Many authors have studied generating functions and their properties and found applications for them for instance, Comtet [ 1 ], Flajolet and Sedgewick [ 2 ], Graham et al. Generating functions have an important role in the study of polynomials. Vast investigations related to the generating functions for many polynomials can be found in many books and articles e.

A special place in this area is occupied by research in the field of obtaining new identities for polynomials and special numbers with using their generating functions.

Interesting results in the field of obtaining new identities for polynomials can be found in some recent works by Simsek [ 181920 ], Kim et al.

Another trend in study of polynomials is getting new representation and explicit formulas for those polynomials. For instance, Qi has recently established explicit formulas for the generalized Motzkin numbers in [ 26 ] and the central Delannoy numbers in [ 27 ]. One can find interesting results in papers of Srivastava [ 2829 ], Cenkci [ 30 ], and Boyadzhiev [ 31 ].

According to Stanley [ 32 ], ordinary generating functions are defined as follows:. Definition 1. Kruchinin et al. Definition 2. Compositae also can be used for calculating the coefficients of generating functions obtained by addition, multiplication, composition, reciprocation, and compositional inversion of generating functions for details see [ 333435 ].

By the reciprocal generating function we mean the following [ 6 ]:. For such generating functions, we obtain several properties, which are given in the following theorem:. Theorem 1. If A t is a generating function of the following form:. Using Eqs. Hence, the composita of Eq. Applying Eq. The generalized Bernoulli polynomials are defined by the following generating function [ 3738 ]:. According to Eq.Polynomials are linear combinations of powers of x.

The highest power is the order of the polynomial. For example, a cubic is a polynomial of order 3. You may be required to manipulate polynomials including expanding brackets, simplifying expressions and factorising. Show Solution. Are you ready to test your Pure Maths knowledge?

Polynomials - linear combinations of powers of x - StudyWell. Manipulating Polynomials Expanding and Simplifying — To expand and simpify — this works in the same way as expanding double brackets but you should end up with 6 terms before simplification.

Factorising Cubics — you may be asked to factorisefor example.

Algebra 84 - Monomial Building Blocks of Polynomial Functions

The trick is to inspect the cubic and see if you can guess a root. It follows that x-1 is a factor.

manipulating polynomials in generalized form

It can be shown that by guessing the quadratic inside the brackets and expanding then improving. See the Cubics pageExample 2. Alternatively, you can use polynomial division as below and seen in Example 3. Fully factorised. It follows from this that if a polynomial is divisible by then and vice versa. For example, if is a factor ofthen. Similarly, if then is a factor. In simpler cases, iffor example, then is a factor of.

Simplify a problem by identifying a factor of a polynomial before you use polynomial division see below.

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Polynomial Division — Polynomial Division combines algebra with the technique of long division. The idea is to identify the factor required for the left most term in each step.

This factor then goes on top. For example, in Example 3, a cubic polynomial with a term is being divided by a linear function with an term. The missing factor is thus. This is then multiplied by to see what remains to find. Examples Example 1 — Manipulating Polynomials Expand and simplify.

manipulating polynomials in generalized form

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Subresultants and Reduced Polynomial Remainder Sequences

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