Degree (mathematics)

Polynomials and fractions

Degree of a polynomial

With unspecified

That is to say $A$ a ring. The ring of the polynomials to unspecified on $A$ is $A$ That is to say $P \ in A$ .

The degree of $P$ , noted $\ deg (P)$ or $d^ \ circ (P)$ is defined by:

If $P=0$ , $\ deg (P) = - \ infty$
If not, for $P = a_n X^n + a_ {N - 1} X^ {N - 1} +… + a_1 X + a_0$ , one defines: $\ deg (P) = \ sup \ {N \ in \ NR, a_ {N} \ 0 \}$

For example, $\ deg (3X^5-2X^4+8X-2) =5$

With several unspecified

A ring and $n \ in \ N$ are $A$ . The ring of the unspecified polynomials to $n$ on $A$ is $A…, X_n$

The degree of the null polynomial is always $- \ infty$ .

If not one considers the whole of the “sums of the exhibitors of unspecified” in each term. The degree of the polynomial is then the largest element of this unit.

For example: in $has, \ deg (X^2 Y^2 + 3X^3 + 4Y) = 4$

Degree of a rational fraction

That is to say $A$ a commutative, unit, just Ring. The body of the rational fractions with unspecified on $A$ is $A (X)$ . Either $F \ in has (X)$ . There exists $N \ in A$ and $D \ in has \ setminus \ {0 \}$ such as $F= \ tfrac {NR} {D}$ .

The size $\ deg (A) - \ deg (B) \ in \ mathbb Z \ cup \ {- \ infty \}$ is independent of the representative $\ tfrac {NR} {D}$ selected for $F$ .

One defines then $\ deg (F) = \ deg (A) - \ deg (B)$ , noted $\ deg (F)$ or $d^ \ circ (F)$ .

Properties of the degree

$\ forall (P, Q) \ in (has (X))^2, \ deg (P+Q) \ Leq \ sup \ {\ deg (P), \ deg (Q) \}$
If $A$ is just, $\ forall (P, Q) \ in (has (X))^2, \ deg (PQ) = \ deg (P) + \ deg (Q)$

Graph and top

In Graph theory, the degree of a top is the number of edges resulting from this top.

See too

Degree of an application

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