...xactly, an extension $L$ of a field $K$ is called the splitting field of a polynomial $f$ over the field $K$ if $f$ decomposes over $L$ into linear factors:
...tsc,a_n)$ (see [[Extension of a field]]). A splitting field exists for any polynomial $f\in K[x]$, and it is defined uniquely up to an isomorphism that is the id
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''minimum polynomial of a matrix''
...l|characteristic polynomial]] of $A$ and, more generally, it divides every polynomial $f$ such that $f(A)=0$.
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...g $w_1=w_2=w$, then the resulting polynomial is called the simple matching polynomial of $G$.
...e been investigated [[#References|[a8]]]. The analytical properties of the polynomial have also been investigated [[#References|[a10]]].
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$#C+1 = 209 : ~/encyclopedia/old_files/data/P073/P.0703700 Polynomial and exponential growth in groups and algebras
is of polynomial growth, or power growth, $ r $
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A divisor of a polynomial $A(x)$ is a polynomial $B(x)$ that divides $A(x)$ without remainder (cf. [[Division|Division]]).
...e of $a$, then $a$ is ''irreducible''. For polynomials, see [[Irreducible polynomial]]; for integers, the traditional terminology is [[prime number]].
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...gebraic number]]. An algebraic irrationality is the root of an irreducible polynomial of a degree at least two, with rational coefficients.
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be a [[Polynomial|polynomial]] of degree $ \geq 1 $
is irreducible (cf. [[Irreducible polynomial|Irreducible polynomial]]) and, trivially, that the leading coefficient is positive. Are these cond
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$#C+1 = 17 : ~/encyclopedia/old_files/data/A011/A.0101620 Algebraic polynomial of best approximation
A polynomial deviating least from a given function. More precisely, let a measurable fun
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A method for calculating the roots of a polynomial
...mials of degree 3. The parabola method allows one to find all zeros of the polynomial without preliminary information about initial approximations. The convergen
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...ferential equations; it is an analogue of the [[Hilbert polynomial|Hilbert polynomial]].
There exists (see [[#References|[2]]]) a polynomial whose value at points $ s \in \mathbf Z $
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A form in four variables, that is, a homogeneous polynomial (cf. [[Homogeneous function|Homogeneous function]]) in four unknowns with c
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A conjecture on the asymptotic behaviour of a polynomial satisfying the Bunyakovskii condition (cf. also [[Bunyakovskii conjecture|B
be polynomials (cf. [[Polynomial|Polynomial]]) with integer coefficients, of degrees $ d _ {1} \dots d _ {r} \geq 1 $
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''polynomial deviating least from zero''
An algebraic polynomial of degree $n$, with leading coefficient 1, having minimal norm in the space
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...next to the knot $8_9$ is written $7-5+3-1$. This means that the Alexander polynomial equals $\Delta(t)=-t^6+3t^5-5t^4+7t^3-5t^2+3t-1$. Non-alternating knots are
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...l and its generalizations (e.g. the [[Jones–Conway polynomial|Jones–Conway polynomial]]).
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...the identity matrix. The multiplicity of an eigen value as a root of this polynomial is called its algebraic multiplicity. For any linear transformation of a fi
...ield $k$ (or a characteristic root of $A$) is a root of its characteristic polynomial.
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''separable game, polynomial-like game''
the degenerate game is called a polynomial game. In any two-person zero-sum degenerate game on the unit square player
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...is unipotent if and only if its [[Characteristic polynomial|characteristic polynomial]] is $ ( x - 1) ^ {n} $.
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...en over all black regions. The second Listing polynomial, or white Listing polynomial, $P _ { W } ( \delta , \lambda )$ is defined in a similar manner, summing o
...corners (e.g. to define the [[Kauffman bracket polynomial|Kauffman bracket polynomial]]), studying labelling of corners of alternating diagrams (e.g. to proof th
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...of a polynomial by a linear binomial: The remainder of the division of the polynomial
...e of Bezout's theorem is the following: A number $\alpha$ is a root of the polynomial $f(x)$ if and only if $f(x)$ is divisible by the binomial $x-\alpha$ withou
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A field $K$ for which every homogeneous polynomial form over $K$ of degree $d$ in $n$ variables with $n > d$ has a non-trivial
...strongly quasi-algebraically closed'' if the same properties holds for all polynomial forms. More generally, a field is $C_i$ if every form with $n > d^i$ has a
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...pt of a limit transition), and makes sense for any coefficient ring. For a polynomial
of a polynomial, then $ x _ {0} $
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...n )$-matrices $A$ such that the [[Characteristic polynomial|characteristic polynomial]] of $A$, $\operatorname { det } ( \lambda I - A )$, is equal to $f$. Indee
...and their minimal polynomial (cf. [[Minimal polynomial of a matrix|Minimal polynomial of a matrix]]) is $f$, i.e. their similarity invariants are $1 , \dots , f$
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A homogeneous polynomial of the first degree (cf. [[Homogeneous function]]).
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...ements for the variables. The ''standard polynomial'' of degree $n$ is the polynomial
...commutative ring satisfies the standard polynomial of degree $2n$, and no polynomial of lower degree.
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