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Thepis polynomial : <=>p:N->R/\ (existskinN:foralli>=k:p_{i}= 0).

We denote by "Poly" the set of all polynomials over the reals:

deg( p) :=ifforalliinN:p_{i}= 0then0else(suchkinN:p_{k}!= 0 /\ (foralli>k:p_{i}= 0)).

Poly := {pinN->R:pis polynomial}.

We then can define the following "number-like" operations on polynomials.

. _{Poly}:R-> Polyc_{Poly}:=suchpin Poly:p_{0}=c/\ (foralli> 0:p_{i}= 0)x:=suchpin Poly:p_{0}= 0 /\p_{1}= 1 /\ (foralli> 1:p_{i}= 0)+: Poly x Poly -> Poly ( p+q)_{i}:=p_{i}+q_{i}-: Poly x Poly -> Poly ( p-q)_{i}:=p_{i}-q_{i}-: Poly -> Poly (- p)_{i}:= -(p_{i})*: Poly x Poly -> Poly ( p*q)_{i}:= (sum_{0 <= j <= i}p_{j}*q_{i-j})

One can show that the polynomial operations satisfy the laws that also hold
for the corresponding operations on numbers, e.g., p*1_{Poly} = p
and p*q = q*p.

The first operation maps a real number `c` in a unique way to a
polynomial `c`_{Poly}. Usually we write just `c` instead
of `c`_{Poly}, when the meaning is clear from the context,
e.g., if `p` is a polynomial, then `p`+1 actually denotes
`p`+1_{Poly}.

3 _{Poly}= [3, 0, 0, 0, 0, ...] x= [0, 1, 0, 0, 0, ...] x+3= [3, 1, 0, 0, 0, ...] x*x= [0, 0, 1, 0, 0, ...] x*x+2*x+1= [1, 2, 1, 0, 0, ...] ( x+1)*(x+2)= [2, 3, 1, 0, 0, ...]

The polynomial operations are closely related to their counterparts on **R**.

a_{Poly}+b_{Poly}= ( a+_{R}b)_{Poly},a_{Poly}-b_{Poly}= ( a-_{R}b)_{Poly},- a_{Poly}= (- _{R}a)_{Poly},a_{Poly}*b_{Poly}= ( a*_{R}b)_{Poly}.

In other words, a property like 1+1=2 also holds for polynomials
1_{Poly} and 2_{Poly} and + interpreted as the
polynomial addition. Above law says that **R** can be embedded into "Poly"
in a sense that is discussed in Section *Embedding Sets*.

Definition *Polynomial Operations* answers the question what a
polynomial __x__+1 is: it is the sum of two polynomials __x__ and
__1__, i.e., the "variable" __x__ in a polynomial is nothing but
a particular (*constant*) polynomial! However, the view of __x__ as
a variable that can be substituted by a value is provided by the following
operation.

[ ]: Poly x R->Rp[a] := (sum_{0 <= i <= deg(p)}p_{i}*a^{i}).

p[5] = 2*5^{0}+3*5^{1}+4*5^{2}= 117.

The evaluation operation satisfies the following laws.

c_{Poly}[a]= c_{Poly},x[a]= a_{Poly},( p+q)[a]= p[a] +q[a],( p*q)[a]= p[a] *q[a].

When evaluating a polynomial `p` on a real number
`a`, we thus just need to substitute `a`_{Poly} for every
occurence of __x__ in the term describing `p` and then evaluate
polynomial addition and multiplication. We can do this by simply using the
rules for addition and multiplication over the real numbers.

( x+1)[2]= 2+1 (= 3 _{Poly})( x*x)[2]= 2*2 (= 4 _{Poly})( x*x+2*x+1)[3]= 3*3+2*3+1 (= 16 _{Poly})

In mathematical practice, we usually do not work with a fixed "polynomial
variable" __x__. Instead we rather introduce a set **R**[`x`] of
polynomials with coefficients in **R** (and correspondingly for other
coefficent domains) such that the constant `x` denotes the polynomial
[0, 1, 0, 0, 0, ...].

[1, 2, 0, 0, 0, ...].

A multivariate polynomial is interpreted as a univariate polynomial whose coefficents are themselves polynomials.

which can be written asx^{2}y+xy^{2}+3x+2y+1

is an element ofy*x^{2}+(y^{2}+3)*x+(2y+1)

[2The coefficient 2y+1,y^{2}+3,y, 0, 0, 0, ...].

[1, 2, 0, 0, 0, ...].

Author: Wolfgang Schreiner

Last Modification: October 4, 1999