Why does is a polynomial in of degree?

Let $P(X)$ be a polynomial over $\mathbb{Z}$ of degree $d-1$ and $n_0$ be some constant positive integer. Then why does $\sum_{k=n_0}^n{P(k)}$ is a polynomial in $n$ of degree $d$?

group theory - Is the complement of a degree 2 divisor in $\mathbb{P}^1_K$ an al

Let $K$ be a field, and let $D\subset\mathbb{P}^1_K$ be an effective divisor of degree 2, defined over $K$. The complement $C := \mathbb{P}^1_K - D$ is thus a smooth curve over $K$, which over the

complex analysis - Degree of $P$ as a smooth function equals its polynomial degr

Let $M,N$ be connected oriented manifolds such that $\partial M=\partial N=\emptyset$. Let $F:M\to N $ be a smooth proper map (i.e. for every $K\subset N$ compact, $F^{-1}(K)$ is compact). We defin...

abstract algebra - Nonsingular projective variety of degree $d$ - Mathematics St

For each $d>0$ and $p=0$ or $p$ prime find a nonsingular curve in $\mathbb{P}^{2}$ of degree $d$. I'm very close just stuck on one small case. If $p\nmid d$ then $x^{d}+y^{d}+z^{d}$ works. If ...

combinatorics - Induction on the degree of a Polynomial. - Mathematics Stack Exc

Let $p(k)$ be a polynomial of degree $d$. Prove $q(n) = \sum_{k=1}^np(k)$ is a polynomial of degree $d + 1$ and that $q(0) = 0$. First I'll prove that it is equivalent to proving degree of $f_d(n)=...

algorithms - If $p(x)$ is a polynomial of degree d, prove that $p(x) \in \Theta(

I just started learning asymptotic notation and I have a problem with this one. Let $p(x)=a_dx^d+a_{d-1}x^{d-1}+.....+a_1x+a_0$ be a polynomial of degree d, with $a_i \in \mathbb{R}$ for $0\leq i ...

If the degree of a divisor on a Riemann surface is $\deg(D) \geq 2g$, then $L(D-

Let $S$ be a compact connected Riemann surface, $D$ a divisor on $S$, and $p\in S$ a point. I want to show that if $\deg(D) \geq 2g$ (where $g$ is the genus of $S$), then we have a strict inclusion

calculus - value of $P''(1 in a degree $4$ of $P(x - Mathematics Stack Excha

If $P(x)$ is a polynomial of degree $4$ with $P(2)=-1,P'(2)=0,P''(2)=2,P'''(2)=-12,P''''(2)=24.$ Then $P''(1)=$ What i try Let $P(x)=ax^4+bx^3+cx^2+dx+e$ So $P'(x)=4ax^3+3bx^2+2cx+d$ And so $P'...

abstract algebra - Prove that a polynomial of degree $d$ has at most $d$ roots (

Let $p(x)$ be a non-zero polynomial in $F[x]$, $F$ a field, of degree $d$. Then $p(x)$ has at most $d$ distinct roots in $F$. Is it possible to prove this without using induction on degree? If so,...

abstract algebra - Does there always exist an irreducible polynomial of degree $

Let $p$ be a prime and let $d$ be a positive integer. Does there always exist an irreducible (i.e. unfactorable) polynomial of degree $d$ over $\mathbb{Z}/p\mathbb{Z}$?