1
vote
0answers
14 views

Self-contained formal polynomial reference

In the forward to the third edition of his Undergraduate Algebra, Lang mentions: A new section in Chapter IV gives a complete account of the Mason-Stothers theorem about polynomials, with Noah ...
0
votes
0answers
11 views

Reading on Laurent Polynomials

I'm interested in reading about Laurent Polynomials. Does anyone know a good resource/book that I can read about Laurent polynomials? Thanks.
1
vote
2answers
48 views

roots of cubic polynomial

On page 26 of Milne's Elliptic Curves (http://www.jmilne.org/math/Books/ectext5.pdf), he states the following: "... a cubic polynomial $h(x) \in k[x]$ with two roots in $k$ has all of its roots in ...
1
vote
1answer
35 views

Lower bound for $(x + y)^k $?

I'm wondering, is there a lower bound for $(x + y)^k $? For example, if $x,y,k > 0$, can we say that $(x + y)^k \geq x^k + y^k$? If anyone has a source/reference for this, that would be great.
2
votes
1answer
64 views

Solving polynomial equations over finite fields

I have looked (a bit) at questions like finding the number of roots of $x^n =1$ over a finite field. Now I would like to understand how to solve polynomial equations over finite fields. From what I ...
0
votes
1answer
79 views

$­\prod_{k=1}^{n}(x_k-a)(x_k-b)\leqslant\sum_{q=1}^{n}x_q^2\prod_{p=1,p\neq q}^{n}(x_p-a)(x_p-b).$?

Is there a name for this formula? For $f_k,w_k\geqslant0$. $$­\prod_{k=1}^{n}f_k\leqslant\sum_{q=1}^{n}w_q\prod_{p=1,p\neq q}^{n}f_p.$$ I believe that there is $w_k$ that make the formula true. Am I ...
0
votes
1answer
54 views

Powers generate monomials

What is a reference in the literature for the following fact? Let $A$ be a commutative $\mathbb{Q}$-algebra. Then every monomial in $A$ of degree $n$ may be written as a linear combination of $n$th ...
0
votes
1answer
77 views

Have you seen this theorem before? (GCD divides, neccessary & sufficient condition)

Conjecture. Let $a,b, c\in \Bbb{Z}, b \neq 0$, The following conditions are equivalent: (1) $d = \gcd(a,b)$ divides c. (2) There's a polynomial in $f \in \Bbb{Z}[X,Y]$ with $c$ constant term, such ...
1
vote
1answer
59 views

A theorem about ideals of $K[T_1,\ldots,T_n]$ and their generators

Suppose that $L\subseteq K$ is a field extension ( we are in characteristic $0$) and moreover that $\mathfrak a\subseteq K[T_1,\ldots,T_n]$ is an ideal ($T_1,\ldots,T_n$ are indeterminates). I have ...
1
vote
0answers
34 views

When is the dot product of roots of certain multivariate polynomials also a root?

Problem. Let $n\in\mathbb N$ be fixed, and suppose that we are given three collections $$z_1,\ldots,z_n\in\mathbb Z,~a_1,\ldots,a_n\in\mathbb R,\text{ and }b_1,\ldots,b_n\in\mathbb R.$$ Suppose we ...
0
votes
1answer
29 views

Zeros of polynomials and power series

Consider $k_i \in \{-1,1\}$ for every $i \in \mathbb{N}$ and consider the family of polynomials $P$ of the form $$\mathop{\sum}\limits_{i=0}^n k_it^i;$$ and the family of power series $S$ of the form ...
0
votes
0answers
78 views

Multivariable irreducible polynomials over finite fields

It is not difficult to prove the following result, and it seems that it should be already proved. I would appreciate it if someone offer me some reference to it. For any $f(x_1,\dots, x_n)=\sum ...
2
votes
1answer
131 views

Degree of a function

I found on wikipedia (http://en.wikipedia.org/wiki/Degree_of_a_polynomial) that a degree of a general function can be computed as $$\deg f(x) = \lim_{x\to\infty}\frac{\log |f(x)|}{\log x}$$ or $$\deg ...
2
votes
1answer
39 views

Help with Polynomial Roots Problem

Let's consider the case of two variables, $p\in\mathbb{R}[x,y]$. Suppose I want to find when there is $c\in\mathbb{R}$ such that $$p(x,x)+p(x,c-x)-p(c-x,x)-p(c-x,c-x)=0 \textbf{ for all } ...
1
vote
1answer
42 views

A property of polynomials in a paper by Rice

It has been suggested that a mind-reading tag be added. This is, unfortunately, a good candidate for such a tag... I was reading a paper of Rice relating to a property of integer polynomials. The ...
12
votes
1answer
230 views

Semialgebraic conditions that convey properties of Galois group

Let $f \in \mathbb{Z}[x]$ be a polynomial of degree $n$ with integer coefficients and let $G_f$ be the Galois group of $f$ over $\mathbb{Q}$. I am trying to collect results that convey some ...
1
vote
0answers
73 views

Orthogonality of the Hermite polynomials: probabilistic approach

Can anyone help me with the following question: Is there any reference in which a probabilistic approach was used to prove that the Hermite polynomials are orthogonal? Thanks a lot!
2
votes
1answer
599 views

History of polynomial arithmetic

How did the notions of polynomial addition,multiplication and division develop historically? The fact that this correspondence with the integers exists seems to be of great importance and is not at ...
2
votes
1answer
87 views

Minimal Polynomial of $\alpha^2$

Having already proved that $p(x)=x^5 + x^2 + 1$ is primitive in $GF(2)$ and assuming that $\alpha$ is a primitive element representing a root of $p(x)$, I am trying to minimal polynomial of $\alpha^2$ ...
3
votes
1answer
70 views

Mutual dependency of polynomial expressions

Suppose you are given the values of $m$ polynomial expressions in $n$ variables. That is we know that $P_1(x_1,x_2,...,x_n)=a_1,P_2(x_1,x_2,...,x_n)=a_2,...,P_m(x_1,x_2,...,x_n)=a_m$ for some ...
5
votes
1answer
136 views

Polynomials in nature

What polynomials occur in "nature"? I am interested in polynomials of degree three and higher. I am aware of Stefan Boltzmann Law and Chemical Equilibrium Examples.
3
votes
1answer
173 views

Polynomials over non-commutative rings

What would be a good source for polynomial expressions over non-commutative rings, such that the variable wouldn't have to commute with the coefficients, so that the substitution of a value from the ...
10
votes
2answers
101 views

Behavior of zeros of $f'$ for complex polynomials $f$ with zeros on the boundary of the unit disc.

Suppose we have $f(z) = (z-r_1)\cdots(z-r_n)$, $|r_j| = 1$. According to the Lucas-Gauss theorem, all of the zeros of $f'$ lie in the convex hull of the $r_j$, but I discovered some behavior I don't ...
2
votes
2answers
91 views

Characterization of polynomial injection from Q to Q?

I want to know if we can find (or characterize) all the polynomials $f(x) \in \mathbb{Q}[x]$ that induces an injection $f : \mathbb{Q} \rightarrow \mathbb{Q}$ by evaluation. Some examples are $x, ...
2
votes
1answer
83 views

Reference request: Newton-Kantorovich hypothesis for polynomials of integral coefficients

Kantorovich's theorem states that the Newton method for finding the roots of a nonlinear function is guaranteed to converge if a parameter $h$, determined by the values of the function and its ...
2
votes
1answer
60 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 ...
2
votes
3answers
116 views

“Interpolation” of polynomials

I'm dealing with a probability problem and I have to understand the following operation on polynomials: let $F$ and $G$ be any two polynomials of variable $p\in [0,1]$ (to be thought of as a Bernoulli ...
3
votes
2answers
142 views

Nice exercises on resultants

I would like to ask if some one knows a source (a book, or lecture notes ect) that contains several nice exercises on resultants of polynomials (it would be nice if there were some solutions as well ...
11
votes
4answers
699 views

Approximating continuous functions with polynomials

Given $\epsilon \gt 0$ and $f \in C^{0}[0,1]$, Weierstrass says that I can find at least one (how many? probably a lot?) polynomial $P$ which approximates f uniformly: $$\sup_{x \in [0,1]} |f(x) - ...
4
votes
1answer
131 views

A “known” polynomial sequence?

Some published papers and books give the impression that if you write down any infinite sequence of polynomials that follows a simple pattern, one will find that it's named after somebody and has an ...
26
votes
3answers
2k views

Does multiplying polynomials ever decrease the number of terms?

Let $p$ and $q$ be polynomials (maybe in several variables, over a field), and suppose they have $m$ and $n$ non-zero terms respectively. We can assume $m\leq n$. Can it ever happen that the product ...
3
votes
1answer
192 views

On the Jacobian Conjecture

I have been asked to do a work on the Jacobian Conjecture for my master's course. While I am familirized with that conjecture and I understand its implications, I would like to ask you all if there is ...
2
votes
2answers
76 views

Are there Graph Polynomials?

Knots can be represented by polynomials like the Jones polynomial. Is there a comparable representation for graphs? How does it work for subclasses like planar, k-regular...? Google doesn't really ...
2
votes
1answer
85 views

Describe invariant polynomials under action of commutative group of order eight.

I believe the question below should be fairly standard in invariant theory ; I hope someone more familiar with it than me can explain a bit more or point to a reference. Let $F$ be polynomial field ...
1
vote
0answers
136 views

Matiyasevich polynomial proof

Can someone provide a proof, or a link to a proof, of why does the Matiyasevich polynomial always generate primes for the nonnegative results? Any help will be appreciated.
2
votes
3answers
128 views

Polynomials as concrete structures

Motivation The structuralist point of view on mathematical objects has two aspects: On the one side, a mathematical object is seen as a concrete structure of abstract dots, e.g. a graph. On the ...
0
votes
1answer
55 views

Polynomial behavior on hyperbolic plane

More a reference request / more information. I was reading some websites about hyperbolic geometry and got to thinking about how would polynomials $(x^2-2)$ behave in such a geometry. So, I need ...
3
votes
1answer
302 views

Obtaining the discriminant of the characteristic polynomial directly from the matrix

Let $M \in \mathbb{Z}_{n \times n}$ be a square matrix with integer coefficients. Let $P(x)$ be its characteristic polynomial $$ P(x) = \det\left(x \cdot \mathbb{I}_{n \times n}- M\right) $$ I ...
1
vote
0answers
161 views

Chebyshev Equioscillation Theorem in $L_{\infty}[a,b]$?

Let $a,b\in\mathbb{R}$, $a<b$. Consider \begin{align} C[a,b] & :=\{f\in\mathbb{R}^{[a,b]}:f\text{ is continuous}\}\text{,} \\ L_{\infty}[a,b] & :=\{f\in\mathbb{R}^{[a,b]}:f\text{ is ...
2
votes
2answers
310 views

Correct order of books for a beginner

what should be the order of the books in which a beginner should do the following books in algebra: -1.E.J. Barbeau POLYNOMIALS -2. Polynomials and Polynomial Inequalities (Graduate Texts in ...
1
vote
0answers
138 views

A special factorization

Suppose that monic polynomial $f(x)\in\Bbb Z[x]$ such that for all $m\in\Bbb Z$, $m>1$, there's no integers $\langle r,r_1,\ldots,r_m\rangle$ such that $f(r)=f(r_1)\cdots f(r_m)$. Is there any ...
3
votes
0answers
71 views

A question that relates polynomials with real roots to infinite sums

What are the known ways that allow us in some cases to precisely calculate the real roots of some specific polynomials by using infinite sums? I'd appreciate any reference on this topic.
13
votes
1answer
419 views

Polynomials all of whose roots are rational

I have two questions about the class of integer-coefficient polynomials all of whose roots are rational. Q1. Is there some way to recognize such a polynomial from its coefficients $a_0, a_1, \ldots, ...
5
votes
0answers
114 views

relationship between solution of quintic in terms of $_{4}F_{3}$ hypergeometric function and theta functions

There is one approach (Bring radical/method of differential resolvents) to the general solution to the quintic that gives the solution for a particular root $v\in\{v_{1},v_{2},v_{3},v_{4},v_{5}\}$ in ...
2
votes
1answer
187 views

General bound on a polynomial's root with the largest norm

Is there a general bound on a polynomial's root with the largest norm? When Rouche's theorem is used, it still seems that the polynomial's root with the largest norm still needs to be found if we ...
10
votes
1answer
404 views

Reference request: Abel or Ruffini's proof of the Abel-Ruffini theorem

The standard proof of the Abel-Ruffini theorem that people learn is based on Galois theory and the notion of a solvable group, but my understanding is that the original proof predates Galois theory. ...
7
votes
1answer
329 views

Holomorphic function of a matrix

A statement is made below. The questions are: (a) Is the statement true? (b) If it is, does it appear in the literature? Here is the statement. For any matrix $A$ in $M_n(\mathbb C)$, write ...
0
votes
0answers
195 views

Multiple perturbations to cubic equation

Suppose $\alpha\in(0,\frac12)$ and $\beta\in(0,\infty)$ are fixed. Initially I have $N\in\mathbb N\backslash\{0\}$ and $n\in\{0,\ldots,N\}$. I'd like to know, as a function $n$, the solution of the ...
1
vote
1answer
53 views

How to construct a polynomial with minimum deviation from zero on the complex region?

I need to compute the analog of Chebyshev polynomials (which give the minimum deviation from zero on [-1,1]) on the given region $\Omega\subset \mathbb C$. More precisely: find $P_n$ such that ...
1
vote
0answers
66 views

What is the most efficient algorithm for constructing an irreducible polynomial?

Theorem: Assuming that the generalized Riemann hypothesis is true, there is a deterministic polynomial time algorithm to find an irreducible polynomial of degree $n$ over $\mathbb{F_p}$ The ...