Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, dividing, factoring and solving for roots.
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1answer
38 views
subproof from theorem of Polya
Suppose we have polynomial:
$$f(z) = z^n + b_{n-1}z^{n-1} + \cdots + b_0$$
It is a complex polynomial of degree $\geq 1$ with leading coefficient $1$. Associate with $f(z)$ set:
$$C := \{z ...
1
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1answer
27 views
Does there exist a degree $n$ polynomial in $n$ variables which vanishes at the vertices of the unit cube?
Let ${\cal C}=\{0,1\}^n$ be the vertices of the unit cube. The polynomial $$ (1-x_1) x_1 x_2 \cdots x_n$$ has degree $n+1$ and vanishes at every point $(x_1, \ldots, x_n) \in {\cal C}$. Does there ...
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6answers
126 views
Proof that if $\forall a f(a) = g(a)$ then $f=g$
How do we prove formally that if:
$\forall a f(a) = g(a)$
$=>$
$f=g$
when
$f,g \in \mathbb F[x]$
5
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0answers
55 views
My proof that if for a k degree polynomial $P(x)$, for the matrix $A$, $P(A)=0$ then $A$ is invertible
Let $P(x)$ be a $k$-degree polynomial with with non-zero free coefficient. Prove that if for matrix $A$, $P(A)$=0, then $A$ is invertible and $A^{-1}$ is $k-1$ degree $A$ polynomial.
Here's my ...
7
votes
1answer
81 views
Killing functions by successive differentiation.
It is clear that a polynomial function $f(x) = a_0 + a_1 x + \dots + a_n x^n$ has the property that some derivative of $f$ vanishes. (Of course, it's the $(n+1)$-th derivative.) One can also check ...
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0answers
77 views
How to show an ideal is zero-dimensional? [duplicate]
Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated by $\{y^2-xy-2xz,y^3+z^2+1, x^2yz-yz\}$. Show that $J$ is zero-dimensional.
How do I go about showing this?
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1answer
68 views
Polynomials maps
Looking at past exam papers I have seen several question of this style but don't know how to answer it. Any help in the methods need to be uses to calculate would be a great help
Consider $\phi: ...
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1answer
38 views
Number of solutions of $P(x_1, …, x_n) = 0$ in $\mathbb{F}_q^n$
I have this exercise :
$q = p^r$ (it is not clearly precised in the exercise that $r = n$). and $\mathbb{F}_q$ is the finite field of cardinal $q$. Let's $P \in \mathbb{F}_q[X_1, ..., X_n]$ of degree ...
2
votes
1answer
56 views
Counting roots of a multivariate polynomial over a finite field
How many roots can there be of a polynomial $f \in K[x_1, x_2, \dots , x_n]$ where $K$ is a finite field and the maximum exponent of $x_i$ in any term is $m$ for all $i$, assuming not all coefficients ...
1
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1answer
19 views
Find $p$ values such that $p$ is prime and $f_p$ is divided by $g$ and $g \cdot x$
Let's consider the following polynomial over $\mathbb Z_p[x]$:
$$f_p = \overline{20}x^4-x^2+\overline{3}$$
I need to:
find the $p$ values so that $p$ is prime and $f_p$ can be divided by
...
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1answer
40 views
find all positive integers for a given diophantine equation involving 4 or 7 variables
Given equation:
Ap + Bq + Cr + Ds + Et + Gu + Vg = K; (Eq. in 7 variables);
suppose we have A, B, C, D initialize with = 1,2,5,10,20,50 and 100 respectively; and K = 50000;
How do we solve it?
...
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4answers
78 views
Finding the value of a polynomial at zero
Given a polynomial $p_n(x)\,$ with $\, n \geq 0 \,$ such that $\, p_0(x)=1 \;,\; p_1(x)=x\;$ and $\; p_n(x)=xp_{n-1}(x)-p_{n-2}(x)\;$ , how can I find $\;p_{10}(0)\; $ ?
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0answers
37 views
Showing $f(x)=g(x^{p^a})$ over field of (prime) characteristic $p>0$.
Let $f$ be a non-constant irreducible polynomial over a field $F$ of (prime) characteristic $p>0$. I need to prove that $f$ can be presented as:
$f(x)=g(x^{p^a})$, where $g$ is irreducible over ...
7
votes
2answers
165 views
Calculate the number of real roots of $x^8-x^5+x^2-x+1 = 0$
Calculate the number of real roots of $x^8-x^5+x^2-x+1 = 0$
My try: $$\left(x^4-\frac{x}{2}\right)^2+\frac{3}{4}x^2-x+1 = ...
1
vote
1answer
28 views
Solving for $f(n+1)$ when $f(k)$ is known for $k=0,1,…,n$
I posted earlier about polynomials but this is different type of problem I think. I seem to have an answer but I mistrust it....
A polynomial $f(x)$ where deg[$f(x)$]$\le{n}$ satisfies $f(k)=2^k$ ...
1
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1answer
31 views
If $f(X) = a_0 + a_1 X + a_2 X^2 \in \mathbb{F}[X]$ then show $f$ is uniquely determined by $f(x)$, $f(y)$, $f(z)$?
This is the exact question:
It's part(ii) that I don't understand - what does it mean and what is it asking me to do? How would I go about constructing a proof? Any help would be much appreciated.
3
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2answers
103 views
finding the remainder of $x^{100}-2x^{51}+1$
I have never been great with polynomials. Here's my problem.
Find the remainder of $f(x)=x^{100}-2x^{51}+1$ when $f$ is divided by $x^2-1$
This sounds easy right? Why can't I figure it out? My ...
1
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0answers
35 views
Upper bound on degree of coefficients required to write polynomials as a linear combination of $f_1,…,f_n$
All polynomials will be elements of $\mathbb{Q}[x]$. Suppose $f_1,...,f_n$ are polynomials of degree at most $d$ which are coprime. What is a (hopefully sharp) upper bound on the degree of ...
2
votes
1answer
48 views
Any comprehensive material to revise the mathematics
I left school long back and so my mathematics knowledge also fades out.
I am trying hard to re-collect the basics about log / permutaion / combination / probability / polynomial equations.
I tried ...
0
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2answers
32 views
Get polynom from polynom, roots of second one are multiplication of first one.
I have a polynomial $P$, with unknown roots $r_1,r_2, ... ,r_n$.
My goal is to find a polynomial $X$ with roots $s_1,s_2, ... ,s_n$, where each $s_i = 2r_i$
I shall get $X$ with no need to find the ...
3
votes
2answers
97 views
Why is the concept of transcendental numbers linked with rational coefficients? Why not real nor complex coefficients?
I've read this:
In mathematics, a transcendental number is a (possibly complex) number
that is not algebraic—that is, it is not a root of a non-zero
polynomial equation with rational ...
2
votes
0answers
46 views
Show that exists $x$ and $y$ such that $P(x)*P(y) < 0 \ ; x,y\in \Bbb C$
I would like to ask how could I perform following proof:
Prove(show) that exists such $x$ and $y$ that $P(x)*P(y) < 0$. Where $P(x)$ and $P(y)$ are polynomials.
$\forall\ x,y\in \Bbb C$
0
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0answers
47 views
Polynomial-Why is the answer for removing parentheses and solving different
I am a 45 yo self-taught software engineer and I am finally studying mathematics. Pretty simple question:
I am using the ALEKS software and during two different lessons on polynomials we get ...
1
vote
2answers
86 views
Roots of polynomials over finite fields
I've been trying to find the decomposition of $x^2-2$ to irreducible polynomials over $\mathbf{F}_5$ and $\mathbf{F}_7$.
I know that for some $a$ in $\mathbf{F}_5$ (for example), $x-a$ divides $x^2-2$ ...
0
votes
2answers
42 views
Bezout's identity in $F[x]$
Let $F$ be a filed and $F[x]$ be a polynomial ring. Let $p(x)$ be an irreducible polynomial show that $\gcd(p(x),q(x)) = 1\Longrightarrow \exists q(x),s(x).\;r(x)p(x)+s(x)q(x) = 1$
I know the proof ...
2
votes
1answer
71 views
how to find bounds on (complex) coefficients from bounds on a polynomial?
I'm trying to prove the following two statements about a polynomial $p$ of degree $n$ with complex coefficients:
If $|p(x)|\le1$ for all real $x$ with $|x|\le1$, then every coefficient of $p$ has ...
0
votes
1answer
63 views
approximating Lipschitz functions by polynomials?
If $f:[0,1]\rightarrow\mathbb{R}$ satisfies $|f(x)-f(y)|<|x-y|\ $ for all $x$, $y\in[0,1]$ and $\epsilon>0$ is fixed, why must there be a polynomial $p$ of degree less than ...
0
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1answer
39 views
Polynomials decomposition into irreduceables
I've been trying to find the composition to irreduceables of the following polynomials with no much success:
X^2 +1 over the field F7
and X^2-2 over the field F5
Is there any method/algorithms I ...
-2
votes
2answers
36 views
Polynomial Form for $f$ a Polynomial Such That $f(1)=0$
What general form, as, for example $ax+by=c$, does the polynomial whose various forms are all evaluation at $1$ to be $0$?
$k_1\overbrace{(x-1)(x+a_1)(x+a_2)\cdots}^{\text{$n$ ...
0
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1answer
17 views
Uniformly discrete on the $\mathbb{Z}[\beta]$
Is the following statement true?
Given $1<\beta<2$ define $\mathbb{Z}[\beta]$ with coefficients belonging to $0$ or $1$, then there exists $\delta$, such that for any $f\,,g\in ...
1
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2answers
37 views
Dimension Recovery of $S \subset P_n(F)$
How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that
$f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset?
2
votes
1answer
66 views
Prove that $D[x]$ is an integral domain if $D$ is one.
Prove if $D$ is an integral domain and $f,g\in D[X]$ are nonzero, then $fg$ does not equal $0$ and $\deg[f(x)g(x)]=\deg f(x) + \deg g(x)$.
I do not know much about this since I just learned about it. ...
1
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1answer
36 views
Lower bound for polynomial with complex coefficient
Let $p(z)=z^{n}+a_{n-1}z^{n-1}+...+a_{1}z+a_{0}$ be a polynomial with complex coefficients. Define $R:=1+\sum_{k=0}^{n-1}|a_k|$. Show that $|p(z)| > R$ for all $z \in \mathbb C$ with $|z|>R$.
...
0
votes
1answer
20 views
Showing a polynomial $f\in\mathbb Q[x]$ is irreducible if it has rational coefficients?
I'm trying to figure out how I can do this for some arbitrary function. Say I find a monic associate of $f$ that we'll call $f_1(x)$. If I then apply Eisenstein's Criterion or Descartes' Rational Root ...
5
votes
1answer
78 views
Does the polynomial $P(x)$ have integer zeros?
The following is a homework question:
Let $P(x)$ be a polynomial with integer coefficients and $P(x_1)=P(x_2)=P(x_3)=P(x_4)=P(x_5)=P(x_6)=P(x_7)=7$ where $x_i$ are distinct integers. Determine if ...
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0answers
28 views
Homogeneity of translated polynomial
I am currently trying to understand the very basics of complex algebraic curves and I came across the following statement in the book by F. Kirwan (Definition 2.9):
The multiplicity of the complex ...
6
votes
1answer
97 views
What is the correspondence between combinatorial problems and the location of the zeroes of polynomials called?
In the wikipedia article on the Italian-born American mathematician and philosopher Gian-Carlo Rota, it is stated that the one combinatorial idea he would like to be remembered for
"... is the ...
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3answers
69 views
Prove that $\gcd(pf,pg) = p \cdot \gcd(f,g)$ when $p,f,g$ are polynomials
How does one start proving this theory?
Prove:
$$\gcd(pf,pg) = p \cdot \gcd(f,g)$$
when
$$p,f,g \in \mathbb F[x] \;,\;\text{The max power multiplier of $p$ is 1 (fixed polynomial)}.$$
0
votes
1answer
35 views
Uniformly convergent sequence of polynomial
Prove or disprove. The limit of a uniformly convergent sequence of polynomial is differentiable.
0
votes
1answer
39 views
Prove that for every positive integer $d$ there exists $C(d)>0$ such that
for every polynomial $p(x)$ with degree $\leq d$, $\max\limits_{x\in[0,1]}|p'(x)| \leq C(d)\max\limits_{x\in [0,1]} |p(x)|$.
There was also a hint given, that says to "use the compactness of a subset ...
2
votes
5answers
62 views
Show polynomial is a Lipschitz function
If $A\subseteq \mathbb{R}$ is a bounded set and $p$ is a polynomial, then show that $p:A\to \mathbb{R}$ is a Lipschitz function.
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2answers
62 views
What is “prime factorisation” of polynomials?
I have the following question:
Find the prime factorisation in $\mathbb{Z}[x]$ of $x^3 - 1, x^4 - 1, x^6 - 1$ and $x^{12} - 1$. You will need to check the irreduciblity in $\mathbb{Z}[x]$, of ...
2
votes
2answers
47 views
Create a formula by given solutions
For my upcoming middle school exams I will need to convert a formula.
I have got the following question:
Create a formula which has the following solutions: $$ x_{1} = 5,\quad x_{2} = -3.$$
The ...
2
votes
1answer
50 views
Subring of Z[x] generated by set of integers and polynomials
Let Z be the ring of integers. We have the subring of Z[x] generated by integers and p1 and p2 (p1 and p2 are polynomials over Z, we note it as Z[p1,p2]). I've got for my homework to investigate if ...
0
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0answers
58 views
Finding the number of negative and the positive zeros
$$f(x)=x^{4} -3x^{3}+5x^{2}-x-2$$
How to find the number of negative and the positive zeros?
2
votes
3answers
31 views
If I have a polynom $p$ with $p(a) = 0$, how to construct a polynom $q$ with $q(a^{-1}) = 0$.
If I have a polynom $p$ which has a field element a as its root, i.e. $p(a) = 0$, how can I construct a polynom $q$ from it with $q(a^{-1}) = 0$. I conjecture that Vieta's formulae might be helpful, ...
0
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0answers
16 views
Understanding the topology of a variety concretely
My ultimate goal is to understand how to compute the cohomology groups of complex algebraic varieties, without having to know what a scheme is.
Therefore I want to be able to handle simple examples, ...
1
vote
3answers
73 views
$\frac{1}{1-x}$ series expansion
How do I know that the expression:
$$\frac{1}{1-x}$$
Is equal to the infinite sum:
$$-\left(\frac{1}{x}\right)-\left(\frac{1}{x}\right)^2-\left(\frac{1}{x}\right)^3-\left(\frac{1}{x}\right)^4+...$$
...
1
vote
1answer
24 views
algorithm for computing polynomial modular two polynomails
I know how to compute polynomial modular another polynomail in polynomial rings.
But what is the fastest algorithm for computing polynomial modular two polynomails in polynomial ring?
For example: ...
1
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2answers
57 views
Is $x^2+1$ irreducible over a cyclotomic field?
Let $K=\mathbb{Q}[\omega]$, where $1+\omega+\omega^2=0$, let $f(X)=X^2+1$. How can i prove irreducibility of $f$ over $K$?
