Multinomial expression

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The multinomial expression (let's cook it, and come British: poly­nomial) means a majority; is rare: Luo who means poly- and a part: -Or nomen is rare: In the word that merged nomós, it is an important mathematical concept to be of the algebra only from the fixed number and a variable or the unsettled original sum and product. I played a big role in modern algebraic establishment historically.

The multinomial expression having one origin of uncertainty has a shape of 3^x3-7x2 + 2x-23. I call "3x3", "-7x2", "2x", "-23" a clause (term to ask for) for all parts. Binomial equation (binomial), Clause 3 expression (trinomial) add Latin distribution numeral to -nomial, and they are invited to the ceremony made only from one clause equally monomial expression (monomial). In other words, I have "a clause" with the multinomial expression "a lot". The use of the word of the binomial equation is slightly rarer than the word of the monomial expression being a frequent appearance and the use of the word for the number of the clauses the Clause 3 expression or more is extremely rare and tends to treat it as a multinomial expression for a mouthful, and there is the thing that is why to classify only monomial expressions in from a multinomial expression exclusively. In addition, there is the style to call a multinomial expression an integral expression (integral expression) [^{explanatory note 1}].

「Description of the beginning class of the multinomial expression（French versionI refer to)

The equation to be given as an equation between multinomial expressions is called a multinomial expression equation, and, in the case of a rational number coefficient, I put in particular it, and it is said with an algebraic equation. The multinomial expression equation describes the point zero of the multinomial expression function.

Table of contents

Number of the complete changes multinomial expression

For an integer non-negative in former (variable) n unsettled in x a₀, a₁, ..., a_n for the fixed number such as a real number of n +1 or the complex number

${\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}}$ 

I name an expression expressed by の form generally with a multinomial expression of 1 yuan (univariate) or number of the complete changes (univariable).

• The f(x) = a_n^xn + a_n-1^xn-1 +⋯+ a₁x + a₀ distance. I call maximum m becoming a_m ≠ 0 then with the degree of this multinomial expression and express it with deg f.
• I call each a_ixⁱ clause of this multinomial expression or item monomial expression and call i the degree of the clause. Or I express the clause of the i next of this multinomial expression for wind to be a_ixⁱ.
  • I call the zeroth clause a₀ a constant term (ていすうこう, constant term, constant). I can consider the mere fixed number to be a multinomial expression having only a constant term. The degree of the multinomial expression only consisting of the constant terms that are not 0 from the definition of the degree is 0. However, I am not defined, or the degree is often defined as - ∞ conveniently when I consider fixed number 0 to be a multinomial expression.
• I call each fixed number a_i a scalar or this multinomial expression coefficient. Particularly, I call am (m = deg f) most highly advanced this multinomial expression coefficient or an item head coefficient (leading coefficient). The most highly advanced coefficient calls 1 multinomial expression single multinomial expression or モニック multinomial expression.

The sign that the multinomial expression expresses a grand total ∑
Use を

${\displaystyle \sum _{k=0}^{n}a_{k}x^{k}}$ 

It is written down とも. x⁰ is 1 as the multinomial expression (monomial expression) then.

The set to belong of the coefficient expresses the whole of the multinomial expression to assume x which is K an origin of uncertainty in K[x]. For example, the whole of the multinomial expression of the real number coefficient expresses R[x], the whole of the multinomial expression of the complex number coefficient with C[x]. There is usually many that it is a defined algebraic system of the addition, subtraction, multiplication and division operation, and set K of the coefficient will assume the thing which can perform the addition, subtraction, multiplication and division operation called the body in particular freely. About a little general (I may not have the identity element which is not necessarily commutation) ring R, a coefficient baggage multinomial expression is defined in it.

I refer to "the form multinomial expression"

For ring R, I prepare new unsettled former xⁿ for unsettled former x and any a non-negative integer n. But x¹ identifies it with x naturally. Increase it the fixed number of unsettled original 冪 xⁿ times;, in other words, is Rn = {axⁿ | I call the cause of 0 ≠ a ∈ R} the monomial expression of the n next. Formal alignment combination of the monomial expression that I take n ∈ N suitable then, and there is

${\displaystyle \sum _{i=0}^{n}a_{i}x^{i}=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x^{1}+a_{0}x^{0}}$ 

On R which assumes x an origin of uncertainty in (for all i a_i ∈ R) call it a multinomial expression (or by a coefficient R baggage). But 0 xⁱ identifies it with 0. I express a meeting to accomplish of the whole multinomial expression on R which assumes x an origin of uncertainty with R[x] and call R with a coefficient ring of R[x]. R is buried in R[x] by a↦ax⁰, and I am usually considered to be R ⊂ R[x] by this identification.

Multinomial expression ring

The details refer to "a multinomial expression ring"

In whole R[x] of the multinomial expression on commutation ring R of the unit

${\displaystyle \sum _{i=0}^{m}a_{i}x^{i},\sum _{j=0}^{l}b_{j}x^{j}\in R[x]}$ 

For (m≤l),

Addition

${\displaystyle (\sum _{i=0}^{m}a_{i}x^{i})+(\sum _{j=0}^{l}b_{j}x^{j})=\sum _{k=0}^{l}(a_{k}+b_{k})x^{k}}$ 

I double a constant (I double a scalar)

${\displaystyle c\cdot \sum _{i=0}^{m}a_{i}x^{i}=\sum _{i=0}^{m}ca_{i}x^{i}}$ 

Multiplication

${\displaystyle (\sum _{i=0}^{m}a_{i}x^{i})(\sum _{j=0}^{l}b_{j}x^{j})=\sum _{k=0}^{m+l}(\sum _{i+j=k}a_{i}b_{j})x^{k}}$ 

などの operation is defined. Because the product in particular supposes that ax = xa consists for arbitrary former a of unsettled former x and ring R and is defined so that a distributive law consists, in R[x], it is in multi dimensions ring on R. Of the R coefficient to assume x an origin of uncertainty in this or it is easily said with the multinomial expression ring on ring R a multinomial expression ring (the number of the complete changes). If R is the ring of the unit, multinomial expression ring R[x] is a ring of the units, too, and multinomial expression ring R[x] is a commutation ring if R is commutation. The unit group of the multinomial expression ring is equal to the unit group of coefficient ring R.

It is Tamaki Euclid, and number of the complete changes multinomial expression ring K[x] on body K can define the division with the rest.

「Division of the multinomial expression（English version、French versionI refer to)

Substitution

Multinomial expression on commutation ring K of the unit

${\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}\in K[x]}$ 

Expression to replace variable x with α for において, α ∈ K, and to be provided

${\displaystyle a_{n}\alpha ^{n}+a_{n-1}\alpha ^{n-1}+\cdots +a_{1}\alpha +a_{0}\in K}$ 

I am called を f(α) and the value that writes it down, and performed substitution (だいにゅう, substitute) of x = α in f(x). Value α ∈ K meeting f(α) = 0 is called in particular a root of multinomial expression f(x) or a point zero.

Representation to be decided by substituting α for x for f(x) ∈ K[x] and α ∈ K

${\displaystyle \varphi _{\alpha }\colon K[x]\to K;\,f(x)\mapsto f(\alpha)}$ 

It becomes the rings associate same model from は K[x] to K.

It is commutation ring R, S of the unit and associate same model h: of the meantime generally When R → S is given; is the associate same model for former α of S

${\displaystyle \psi _{h,\alpha }\colon R[X]\to S}$ 

Well, as for the thing becoming ψ_h,α(r) = h(r), one only always exists if it is ψ_h,α(X) = α and r ∈ R. This time, R coefficient multinomial expression

${\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}}$ 

It is

${\displaystyle \psi _{h,\alpha }(f)=h(a_{n})\alpha ^{n}+h(a_{n-1})\alpha ^{n-1}+\cdots +h(a_{1})\alpha +h(a_{0})}$ 

I write it as を f(α) and say the value that substituted α for X.

Multinomial expression function

The details refer to "a multinomial expression function" (French version)

For multinomial expression f(X) = a_n^Xn + a_n-1^Xn-1 +⋯+ a₁X + a₀ ∈ K[X], it is a function by the substitution of the value to variable X

${\displaystyle f\colon K\to K;\ x\mapsto f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}}$ 

But, it is decided. I call such function f a multinomial expression function collectively (defined on K). Particularly, the multinomial expression function that f determines is called n next function when degree deg f of multinomial expression f is n.

• The function to be decided from a multinomial expression of the form of y = ax + b (a, b ∈ K, a ≠ 0) is called a linear function.
• The function to be decided from a multinomial expression of the form of y = ax2 + bx + c (a, b, c ∈ K, a ≠ 0) is called a quadratic function.

If set K of the coefficient is infinite bodies such as real number field R or complex number body C, the different multinomial expression establishes a different function. When K is a general commutation ring, it is not this limit. For example, multinomial expression X2 + X is not 0 in limited body F₂, but the function that this determines is 0. If a degree is smaller than the order of magnitude of the body on in an infinite body or the limited body, such a thing is not known to happen.

As for the differential calculus integral calculus of the multinomial expression, the following expressions are basic:

${\displaystyle {\frac {d}{dx}}\,x^{n}=nx^{n-1},\quad \int x^{n}dx={\frac {x^{n+1}}{n+1}}+C.}$ (${\displaystyle C}$ は constant of integration)

When I consider x to be a variable to take a value to a real number and complex number for analytics, this is a fact led by a definition of differential calculus, the integral calculus for the function. On the other hand, I often handle this ceremony as a definition algebraically (I refer to form differential calculus (English version)).

For example, the differential calculus (導多項式) of multinomial expression x2-3x +1 becomes 2x-3.

• A multinomial expression function of double bare variables is the analysis function that is Masanori in the whole area of the Gaussian plane (整関数).
• It is the equivalent that a multinomial expression having an equal root and the multinomial expression have a common factor between 導多項式 of oneself. The multinomial expression having only a single root is called a separation multinomial expression

Multivariate multinomial expression

The details refer to "a multivariate multinomial expression" (French version)

The multinomial expression of multidimensional (multivariate) or multivariate (multivariable) is defined as follows. Unsettled former x₁, x₂, of the m unit …, x_m and fixed number n₁, n₂, of the m unit …Including the thing which is not 0 of, n_m and at least one (n1 +1) (n2 +1)…(nm +1) fixed number a_{e1e2 of the unit…e}m (0≤e_i≤n_i, for i = 1, 2 …For, m),

${\displaystyle \sum _{(e_{1},e_{2},\ldots ,e_{m})\in \mathbb {N} ^{m} \atop 0\leq e_{m}\leq n_{m}}a_{e_{1}e_{2}\ldots e_{m}}x_{1}^{e_{1}}x_{2}^{e_{2}}\cdots x_{m}^{e_{m}}}$ 

I call an expression shown と to the multinomial expression (m-ary polynomial) of the m variable. For example, when pay attention to x₁; variable x₂, of one m- …I can consider a multinomial expression about, x_m to be a number of the complete changes multinomial expression of x₁ lasting for a coefficient. In other words, it is variable x₁, x₂, on commutation ring R of the unit …Multivariate multinomial expression ring R[x1, x₂, about, x_n …, x_n] a multinomial expression as a coefficient baggage multinomial expression

${\displaystyle R[x_{1},x_{2},\ldots ,x_{n}]:=R[x_{1},x_{2},\ldots ,x_{n-1}][x_{n}]}$ 

I can define のように inductively.

Fixed number a_{e1e2 to double of unsettled original 冪積 x1^e1 x2^e2⋯xm^em…I call e}m x_1^e1 x_2^e2⋯xm^e_m the clause of this multinomial expression. I only call the multinomial expression consisting of one clause a monomial expression.

Introduce the concept of the multiplex index; and α = (e₁, e₂, …For, e_m)

${\displaystyle a_{\alpha }\mathbf {x} ^{\alpha }:=a_{e_{1}e_{2}\ldots e_{m}}x_{1}^{e_{1}}x_{2}^{e_{2}}\cdots x_{m}^{e_{m}}}$ 

I can express a multivariate multinomial expression easily when I decide to promise と.

I can display the multinomial expression of the m variable that I gave you first as follows by letting young figure Y_α support multiplex index α for a combination theory, and introducing suitable order ≤ between young people figures more.

${\displaystyle \sum _{\alpha \in \mathbb {N} ^{m},\,Y_{0}\leq Y_{\alpha }\leq Y}a_{\alpha }\mathbf {x} ^{\alpha}}$ .

But Y₀ "index (0, 0 of" minimum …In, 0) Y "index (n₁, n₂, of" maximum ...It is a young figure supporting, n_m) each. It is natural, but is the same thing even if I do not go through a young figure if I give a direct index order. In other words, α = (a₁, a₂, …, am),β= (b₁, b₂, ...For, b_m) α≤β ⇔ ai≤bi (for all i = 1, 2 …You should promise with, m).

Degree of the multinomial expression

The details refer to "the degree of the monomial expression" and "the degree of the multinomial expression"

Multiplex index α = (a₁, a₂, …For, a_m), it is size of α |α| を

${\displaystyle |\alpha |:=a_{1}+a_{2}+\cdots +a_{m}}$ 

It determines で. Variable x = (x₁, x₂, … For clause a_αx_α about x_n), it is | α| Of item をこの fix it for a degree (about x). Multinomial expression f(x) = ∑
For
α

This article is taken from the Japanese Wikipedia Multinomial expression

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