polynomial

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A polynomial sums the multiples of powers of a variable or indefinite :

${\displaystyle P(x)=a_{0}+a_{1}x+a_{2}x^{2}+\dotsb +a_{n}x^{n},\quad n\geq 0}$

or briefly with the sum symbol :

${\displaystyle P(x)=\sum _{i=0}^{n}a_{i}x^{i},\quad n\geq 0}$

Here is the sum symbol, the numbers are the respective multiples and is the indeterminate. ${\displaystyle \textstyle \sum }$ ${\displaystyle a_{i}}$ ${\displaystyle x}$

Exponents of powers are natural numbers . The sum is also always finite . Infinite sums of multiples of powers with natural-number exponents of an indeterminate are called formal power series .

There are some important special polynomials for math and physics .

In elementary algebra this expression is identified with a function in (a polynomial function ), in abstract algebra a strict distinction is made between this term and that of a polynomial as an element of a polynomial ring . In school mathematics, a polynomial function is also referred to as a completely rational function . ${\displaystyle x}$

This article also explains the mathematical terms: degree of a polynomial, leading coefficient, normalization of a polynomial, polynomial term, absolute term, binomial; as well as zero bound, Cauchy's rule, Newton's rule, even and odd powers.

etymology

The word polynomial means something like "multi-named". It comes from the Greek πολύ polý "much" and όνομα onoma "name". This name goes back to Euclid's elements . In book X he calls a two-part sum ἐκ δύο ὀνομάτων (ek dýo onomátōn): "consisting of two names". The term polynomial goes back to Vieta : In his Isagoge (1591) he uses the expression polynomia magnitudo for a quantity with several terms . ${\displaystyle a+b}$

Polynomials in elementary algebra

Graph of a 5th degree polynomial function

definition

In elementary algebra , a polynomial function is a function of form ${\displaystyle P}$

${\displaystyle P(x)=\sum _{i=0}^{n}a_{i}x^{i}=a_{0}+a_{1}x+a_{2}x^{2}+\dotsb +a_{n-1}x^{n-1}+a_{n}x^{n},\quad n\geq 0}$,

where any - algebra is possible as the domain of definition for the (independent) variable , if the range of values ​​is the coefficients (see below). However, this is often the set of whole , real or complex numbers . ${\displaystyle x}$ ${\displaystyle R}$ ${\displaystyle R}$

• They come from a ring , e.g. B. a field or a residue class ring , and are called coefficients . If the whole , the real or the complex numbers are included, one also speaks of whole , real or complex polynomials . ${\displaystyle a_{i}}$ ${\displaystyle R}$ ${\displaystyle R}$
• All exponents are natural numbers .

• The degree of the polynomial is the highest exponent for which the coefficient of the monomial is not zero. This coefficient is leading coefficient (also: leading coefficient ). (The notation for the degree of the polynomial is derived from the English term degree . The German-language literature also often contains the notation or , which comes from German .) ${\displaystyle i\in \{0,\ldots ,n\}}$ ${\displaystyle a_{i}}$ ${\displaystyle a_{i}x^{i}}$ ${\displaystyle \deg f}$ ${\displaystyle f}$ ${\displaystyle \mathrm {grad} \,f}$ ${\displaystyle \mathrm {Grad} \,f}$
• For the zero polynomial where all are zero, the degree is defined as. ${\displaystyle a_{i}}$ ${\displaystyle -\infty }$
• If the leading coefficient is 1, then the polynomial is called normalized or monic .
• If the coefficients are relatively prime , or if the content is 1, then the polynomial is called primitive.

The coefficient is called the absolute term. is called a linear term , a square term and a cubic term. ${\displaystyle a_{0}}$ ${\displaystyle a_{1}x}$ ${\displaystyle a_{2}x^{2}}$ ${\displaystyle a_{3}x^{3}}$

Simple example

By

${\displaystyle P(x):=9x^{3}+x^{2}+7x-3{,}8}$

a third degree polynomial is given (the highest occurring exponent is 3). In this example, 9 is the leading coefficient (as a factor before the highest power of ), the other coefficients are: 1 ; 7 and −3.8 . ${\displaystyle x}$

Designation of special polynomials

Polynomials of degree

• 0 are called constant functions (e.g. ). ${\displaystyle P(x)=-1}$
• 1 linear functions or more precisely affine linear functions are called (e.g. ). ${\displaystyle P(x)=3x+5}$
• 2 are called quadratic functions (e.g. ). ${\displaystyle P(x)=-3x^{2}-4x+1}$
• 3 are called cubic functions (e.g. ). ${\displaystyle P(x)=4x^{3}-2x^{2}+7x+2}$
• 4 are called quartic functions or biquadratic functions (e.g. ). ${\displaystyle P(x)=6x^{4}-x^{3}+4x^{2}+2x+2}$

Zeroing the polynomial

The zeros of a polynomial function or roots or solutions of a polynomial equation are those values ​​of for which the function value is zero, i.e. i.e., which satisfy the equation . A polynomial over a body (or more generally an integrity ring ) always has at most as many zeros as its degree indicates. ${\displaystyle x}$ ${\displaystyle P(x)}$ ${\displaystyle P(x)=0}$

General properties

• The fundamental theorem of algebra states that a complex polynomial (i.e. a polynomial with complex coefficients) of degree has at least one complex zero (pure existence theorem). Then it has exactly zeros ( polynomial division ) if the zeros are counted according to their multiplicity . For example, the root of the polynomial a double . As a result, every complex polynomial of positive degree can be broken down into a product of linear factors . ${\displaystyle n\geq 1}$ ${\displaystyle n}$ ${\displaystyle x=2}$ ${\displaystyle (x-2)^{2}}$
• In general, one can find an algebraic field extension for every field , in which all polynomials of positive degree with coefficients in decay as polynomials into linear factors. In this case it is called the algebraic closure of . ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle L}$ ${\displaystyle K}$
• Every rational zero of a normalized polynomial (highest coefficient is 1) with integer coefficients is an integer and a divisor of the absolute term, the theorem about rational zeros is more general .
• The zeros of polynomials of the first, second, third and fourth degree can be calculated exactly with formulas (e.g. pq formula for quadratic equations), whereas polynomials of higher degree can only be factored exactly in special cases with the help of radical symbols ( Abel theorem -Ruffini ).
• Polynomials of odd degree with real coefficients always have at least one real zero.

Zero point barriers

The position of all zeros of a polynomial of degree can be estimated by zero bounds, in the calculation of which only the coefficients and the degree of the polynomial are included. ${\displaystyle n}$

Real zero bounds

An important special case are real zero bounds for real polynomials: A number is called a real zero bound of the real polynomial if all real zeros of lie in the interval ; it is called the upper real zero bound of if all real zeros of are less than or equal to. Lower zero point boundaries are explained in the same way. ${\displaystyle B\in \mathbb {R} _{+}}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle [-B,B]}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle B}$

The following are examples of real zero bounds for normalized polynomials ; every polynomial can be converted to this form by division. For some real zero bounds, the sub-index set of the genuinely negative coefficients of plays a special role, denoting their number. ${\displaystyle \textstyle f=X^{n}+\sum _{i=0}^{n-1}a_{i}X^{i}}$ ${\displaystyle N=\left\{k\in \{0,1,\dotsc ,n-1\}\mid a_{k}<0\right\}}$ ${\displaystyle f}$ ${\displaystyle |N|}$

• ${\displaystyle B=\max \left\{\left.\left(|N|\cdot |a_{i}|\right)^{\frac {1}{n-i}}\right|i\in N\right\}}$is an upper real zero bound (Cauchy's rule),
• ${\displaystyle B=\min\{x\in \mathbb {R} :f^{(i)}(x)\geq 0{\text{ für alle }}i=0,\dotsc ,n\}}$is an upper real zero bound (Newton's rule),
• ${\displaystyle B=1+{\sqrt[{n-k}]{\alpha }}}$is an upper real zero bound (rule of Lagrange and Maclaurin) , where denotes the absolute value of the negative coefficient with the greatest magnitude and the exponent of the highest term with the negative coefficient; ${\displaystyle \alpha =\max\{|a_{m}|:a_{m}<0\}}$ ${\displaystyle k=\max\{m:a_{m}<0\}}$
• Each that has the inequality ${\displaystyle B\in \mathbb {R} _{+}}$

${\displaystyle B^{n}\geq \sum _{i=0}^{n-1}|a_{i}|B^{i}}$

fulfilled, is a real zero digit limit (such are even limits for the absolute values ​​of complex zeros of complex polynomials). Special cases of this are (see also Gerschgorin's theorem ) ${\displaystyle B}$

• • ${\displaystyle B=1+\max \nolimits _{i=0,\,\dotsc ,\,n-1}|a_{i}|}$ and
  • ${\displaystyle B=\max \left(1,\sum _{i=0}^{n-1}|a_{i}|\right)}$.

Complex zero barriers

For complex polynomials , circles around the zero point of the complex number plane are common as counterparts to the real zero point bounds, the radius of which is to be selected so large that all (or, depending on the application, only "some") complex zeros of the polynomial on the circular disk with it Radius. A number is called a complex zero point limit of the complex polynomial if all zeros of on the circular disk are around the zero point with a radius (or in other words: if the absolute value of each zero point is less than or equal ). A result for complex polynomials is: ${\displaystyle f}$ ${\displaystyle B\in \mathbb {R} _{+}}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle B}$ ${\displaystyle B}$

• Each that has the inequality ${\displaystyle B\in \mathbb {R} _{+}}$

${\displaystyle |a_{k}|B^{k}\geq \sum _{i\in \{0,\dotsc ,n\}\setminus \{k\}}|a_{i}|B^{i}}$

fulfilled, defines a circle in the complex plane with a radius around the origin, which contains exactly complex zeros (consequence from Rouché's theorem ). This inequality is solvable forever, but not necessary for every index . ${\displaystyle B}$ ${\displaystyle k}$ ${\displaystyle k=0,n}$ ${\displaystyle k=1,\dotsc ,n-1}$

• In this case , the bound already given for real polynomials results for the absolute value of all zeros. All direct calculations of specified there continue to apply. ${\displaystyle k=n}$ ${\displaystyle B}$
• In this case, the result is a circle that does not contain any zeros. is then a bound for all zeros of the "reciprocal" polynomial . ${\displaystyle k=0}$ ${\displaystyle 1/B}$ ${\displaystyle x^{n}f(1/x)/a_{0}}$

Solution formulas

In principle, there are several ways to determine the zeros of a polynomial. General iteration methods , such as the Newton method and the Regula falsi or iteration methods specializing in polynomials, such as the Bairstow method or the Weierstrass (Durand-Kerner) method can be applied to every polynomial, but lose out if there are multiple or closely spaced zeros Accuracy and speed of convergence.

For quadratic equations , cubic equations and quartic equations there are general solution formulas, for polynomials of higher order there are solution formulas, provided they have special forms:

• Reciprocal polynomials have the form

${\displaystyle f(x)=c_{0}\cdot x^{n}+c_{1}\cdot x^{n-1}+\dotsb +c_{1}\cdot x+c_{0}}$

d. H. applies to the -th coefficient ; in other words: the coefficients are symmetrical. For these polynomials and those that meet a slight modification of this symmetry condition, the determination of the zeros with the help of substitution (or ) can be reduced to a polynomial equation whose degree is half as large. See reciprocal polynomial for details . ${\displaystyle i}$ ${\displaystyle c_{i}=c_{n-i}\,}$ ${\displaystyle z=x+1/x}$ ${\displaystyle z=x-1/x}$

• Binomials have the form ${\displaystyle f(x)=x^{n}+c\,}$

If we assume real, then the solutions are multiples of the complex -th roots of unity : ${\displaystyle c}$ ${\displaystyle n}$ ${\displaystyle n}$

${\displaystyle x_{k}={\sqrt[{n}]{c}}\cdot \exp \left({2k\pi \mathrm {i} \over n}\right),\quad c\geq 0}$

${\displaystyle x_{k}={\sqrt[{n}]{\vert c\vert }}\cdot \exp \left({(2k+1)\pi \mathrm {i} \over n}\right),\quad c<0}$,

where runs through. ${\displaystyle k=0,\dotsc ,n-1}$

• Polynomials that contain only even powers of have the form: ${\displaystyle x}$

${\displaystyle f(x)=c_{n}\cdot x^{n}+c_{n-2}\cdot x^{n-2}+c_{n-4}\cdot x^{n-4}+\dotsb +c_{4}\cdot x^{4}+c_{2}\cdot x^{2}+c_{0}}$

The solution occurs through substitution . If a solution for has been found, it must be taken into account that two solutions for can be derived from this: ${\displaystyle z=x^{2}\,}$ ${\displaystyle z_{1}}$ ${\displaystyle x}$

${\displaystyle x_{1}={\sqrt {z_{1}}}}$ and ${\displaystyle x_{2}=-{\sqrt {z_{1}}}}$

• Polynomials that contain only odd powers of have the form: ${\displaystyle x}$

${\displaystyle f(x)=c_{n}\cdot x^{n}+c_{n-2}\cdot x^{n-2}+\dotsb +c_{5}\cdot x^{5}+c_{3}\cdot x^{3}+c_{1}\cdot x}$

Here 0 is obviously a zero of the polynomial. One divides the polynomial by and then treats it like a polynomial -th degree, which only contains even powers of . ${\displaystyle x}$ ${\displaystyle (n-1)}$ ${\displaystyle x}$

Polynomials in linear algebra

• The vector space of all real polynomial functions of arbitrary but finite degree is an example of a vector space in linear algebra , which cannot obviously be illustrated by means of geometric ideas .

• The characteristic polynomial is among others in the diagonalization of matrices calculated and studied.

Polynomials in abstract algebra

→ Main article : Polynomial ring

definition

In abstract algebra , a polynomial is defined as an element of a polynomial ring . This in turn is the extension of the coefficient ring by an indefinite, (algebraically) free element . Thus contains the potencies , and their linear combinations with . These are all elements, i. that is, every polynomial is unique by the sequence ${\displaystyle R[X]}$ ${\displaystyle R}$ ${\displaystyle X}$ ${\displaystyle R[X]}$ ${\displaystyle X^{n}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \textstyle a_{0}+\sum _{k=1}^{n}a_{k}X^{k}}$ ${\displaystyle a_{k}\in R}$

${\displaystyle (a_{0},a_{1},\dots ,a_{n},0,0,\dots )\in R\times R\times R\times \dots }$

characterized by its coefficients.

construction

Conversely, a model of the polynomial ring can be constructed using the set of finite sequences in . For this purpose, an addition “ ” is defined as the sum of the sequences in terms of terms and a multiplication “ ” is defined by convolution of the sequences. So is and , so is ${\displaystyle R[X]}$ ${\displaystyle R\times R\times R\times \dots }$ ${\displaystyle R[X]}$ ${\displaystyle +}$ ${\displaystyle \cdot }$ ${\displaystyle a=(a_{n})_{n\in \mathbb {N} _{0}}}$ ${\displaystyle b=(b_{n})_{n\in \mathbb {N} _{0}}}$

${\displaystyle a+b:=(a_{n}+b_{n})_{n\in \mathbb {N} _{0}}}$

and

${\displaystyle a\cdot b:=\left(\sum _{i=0}^{n}a_{i}b_{n-i}\right)_{n\in \mathbb {N} _{0}}=\left(\sum _{i+j=n}a_{i}b_{j}\right)_{n\in \mathbb {N} _{0}},}$

${\displaystyle R[X]}$with these links there is now itself a commutative ring, the polynomial ring (in an indeterminate) over . ${\displaystyle R}$

If one identifies the indefinite as a sequence , so that , etc., then every sequence can again be represented in the intuitive sense as a polynomial as ${\displaystyle X:=(0,1,0,0,\dotsc )}$ ${\displaystyle X^{2}=X\cdot X=(0,0,1,0,0,\dotsc )}$ ${\displaystyle X^{3}=X^{2}\cdot X=(0,0,0,1,0,0,\dotsc )}$ ${\displaystyle (a_{0},a_{1},a_{2},\dotsc )\in R[X]}$

${\displaystyle (a_{0},a_{1},a_{2},\dotsc )=a_{0}+a_{1}\cdot X+a_{2}\cdot X^{2}+\dotsb =a_{0}+\sum _{n\in \mathbb {N} _{>0}}a_{n}\cdot X^{n}.}$

Relation to the analytical definition

If one now considers that according to the assumption there exists a natural number , so that applies to all , then, according to the above considerations, every polynomial over a commutative unitary ring can be clearly written as . However, this is not a function as in analysis or elementary algebra, but an infinite sequence (an element of the ring ) and is not an “unknown”, but the sequence . However, one can use it as a “pattern” to then form a polynomial function (i.e. a polynomial in the ordinary analytical sense). For this one uses the so-called insertion homomorphism . ${\displaystyle n\in \mathbb {N} _{0}}$ ${\displaystyle a_{i}=0}$ ${\displaystyle i>n}$ ${\displaystyle f\in R[X]}$ ${\displaystyle f=a_{0}+a_{1}\cdot X+\dotsb +a_{n}\cdot X^{n}}$ ${\displaystyle f}$ ${\displaystyle R[X]}$ ${\displaystyle X}$ ${\displaystyle (0,1,0,0,\dotsc )}$ ${\displaystyle f}$

It should be noted, however, that different polynomials can induce the same polynomial function. For example, if the remainder class ring , the polynomials induce ${\displaystyle R}$ ${\displaystyle \mathbb {Z} /3\mathbb {Z} =\{{\bar {0}},{\bar {1}},{\bar {2}}\}}$ ${\displaystyle f,g\in (\mathbb {Z} /3\mathbb {Z} )[X]}$

${\displaystyle f=X(X-{\bar {1}})(X-{\bar {2}})=X^{3}-{\bar {3}}X^{2}+{\bar {2}}X=X^{3}-X}$

and

the zero polynomial ${\displaystyle g=0}$

both the zero mapping , that is: for everyone ${\displaystyle 0\in \operatorname {Abb} \left(\mathbb {Z} /3\mathbb {Z} ,\mathbb {Z} /3\mathbb {Z} \right)}$ ${\displaystyle f(x)=g(x)={\bar {0}}=0(x)}$ ${\displaystyle x\in \mathbb {Z} /3\mathbb {Z} .}$

For polynomials over the real or integers or in general any infinite integrity ring , however, a polynomial is determined by the induced polynomial function.

The set of polynomial functions with values ​​in also forms a ring ( sub-ring of the function ring ), which is only rarely considered. There is a natural ring homomorphism of the ring of the polynomial functions whose core is the set of polynomials which induce the zero function. ${\displaystyle R}$ ${\displaystyle R[X]}$

Generalizations

Polynomials in several indeterminates

In general, one understands every sum of monomials of the form as a polynomial (in several indeterminates): ${\displaystyle a_{i_{1},\dotsc ,i_{n}}X_{1}^{i_{1}}\dotsm X_{n}^{i_{n}}}$

${\displaystyle P(X_{1},\dotsc ,X_{n})=\sum _{i_{1},\dotsc ,i_{n}}a_{i_{1},\dotsc ,i_{n}}X_{1}^{i_{1}}\dotsm X_{n}^{i_{n}}}$

Read: "Capital-p from capital-x-1 to capital-xn (is) equal to the sum over all i-1 up to in from ai-1-bis-in times capital-x-1 to the power of i-1 to capital- xn high in "

With a monomial order it is possible to arrange the monomials in such a polynomial and thereby generalize terms such as guide coefficient .

The size is called the total degree of a monomial . If all (non-vanishing) monomials in a polynomial have the same total degree, it is said to be homogeneous . The maximum total degree of all non-vanishing monomials is the degree of the polynomial. ${\displaystyle i_{1}+\dotsb +i_{n}}$ ${\displaystyle X_{1}^{i_{1}}\dotsm X_{n}^{i_{n}}}$

The maximum number of possible monomials of a given degree is

${\displaystyle {\binom {n+k-1}{k}},}$

Read: "n + k-1 over k" or "k from n + k-1"

where is the number of occurring indeterminates and the degree. A problem of combinations with repetition (replacement) is clearly considered here. ${\displaystyle n}$ ${\displaystyle k}$

If one adds up the number of possible monomials of degree up to , one obtains for the number of possible monomials in a polynomial of a certain degree : ${\displaystyle 0}$ ${\displaystyle k}$

${\displaystyle {\binom {n+k}{k}}}$

Read: "n + k over k" or "k from n + k"

If all indeterminates are in a certain way “equal”, the polynomial is called symmetric . What is meant is: if the polynomial does not change when the indeterminate is swapped.

The polynomials in the indeterminate above the ring also form a polynomial ring, written as . ${\displaystyle n}$ ${\displaystyle X_{1},\dotsc ,X_{n}}$ ${\displaystyle R}$ ${\displaystyle R[X_{1},\dotsc ,X_{n}]}$

Formal power series

Going to infinite rows of form

${\displaystyle f=\sum _{i=0}^{\infty }a_{i}X^{i}}$

Read: "f (is) equal to the sum of i equal to zero to infinity of ai (times) (capital) x to the power of i"

over, one obtains formal power series .

Laurent polynomials and Laurent series

If you also allow negative exponents in a polynomial, you get a Laurent polynomial . Corresponding to the formal power series, formal Laurent series can also be considered. These are objects of form

${\displaystyle f=\sum _{i=-N}^{\infty }a_{i}X^{i}.}$
Read: "f (is) equal to the sum of i equal to minus (capital) n to infinity of a − i (times) (capital) x to the power of i"

Posynomial functions

If you allow several variables and any real powers, you get the concept of the posynomial function .

literature

• Albrecht Beutelspacher: Linear Algebra. 8th edition, ISBN 978-3-658-02413-0 , doi : 10.1007 / 978-3-658-02413-0
• Michael Holz & Detlef Wille: Repetition of Linear Algebra, Part 2 , ISBN 978-3923923427
• Gerd Fischer: Textbook of Algebra , ISBN 978-3-658-02221-1 , doi : 10.1007 / 978-3-658-02221-1

Web links

Wiktionary: Polynomial  - explanations of meanings, word origins, synonyms, translations

• Java applet for calculating the (also complex) zeros of polynomials up to 24th degree (according to Newton's method )

Individual evidence

1. ↑ cf. Barth, Federle, Haller: Algebra 1. Ehrenwirth-Verlag, Munich 1980, p. 187, footnote **, there explanation of the designation "binomial formula"
2. ↑ For the usefulness of this setting, see division with remainder .
3. ↑ Ernst Kunz: Introduction to Algebraic Geometry. P. 213, Vieweg + Teubner, Wiesbaden 1997, ISBN 3-528-07287-3 .