Questions tagged [polynomials]

Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.

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4
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0answers
26 views

Greatest common divisor in $\mathbb{F}_p[T]$ with powers of linear polynomials

Let $n>1$ be a natural number, let $p$ be an odd prime number with $p-1 \mid n-1$ such that $p^k - 1 \mid n-1$ does not hold for any $k>1$. In the following, we work in the ring $\mathbb{F}_p[T]$...
3
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2answers
182 views

Largest absolute value of a polynomial of degree $n$ on $\{0,1,\ldots,n\}$

Consider a polynomial $P_n(x)\in\mathbb{R}[x]$, of degree $n\geq1$, of the form $$P_n(x)=c_0+c_1x+c_2x^2+\cdots+c_{n-1}x^{n-1}+x^n.$$ To illustrate the question, take $P_1(x)=c_0+x$ so that $P_1(0)=...
11
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2answers
202 views

Curious identity between the two kinds of Chebyshev polynomials

I have found, by accident, an identity that relates a sum of Chebyshev polynomials of the first kind to a Chebyshev polynomial of the second kind. It goes as follows: Given an integer partition of $n$...
-1
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0answers
23 views

solution of a variable complex polynomial

Let $P(z)= (2+\alpha)z^q + \alpha z^m - \alpha z^{q-m} +2-\alpha$ where: $\alpha > 0$; q>m and $m=2n+1$ Let $Z(P)=\lbrace z\in\mathbb{C} ; P(z)=0 \rbrace $ and let $\phi(\alpha)=$ max$|z|; z\in Z(P)...
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21 views

Name for product of square-free content of invariant factors of a matrix

Let $M(\lambda)$ be a possibly non-square polynomial matrix (over $\mathbb{R}$ or $\mathbb{C}$ is sufficient for me, but could be more general). By standard theory, it can be put into Smith normal ...
3
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77 views

Special irreducible polynomials in $k[x,y]$

The following question I have asked in MSE, getting one comment. Hopefully, it is ok to ask it here also. Let $k$ be a field of characteristic zero, $n \in \mathbb{N}$. Definitions: (1) $0 \neq f \...
2
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34 views

When flatness of $S$ over $R_i$ implies flatness of $S$ over the ring generated by $R_1,R_2$

The following question I have asked in MSE, but have not received an answer, so I ask it here; I really apologize if it is not suitable for MO. Let $k$ be a field of characteristic zero and let $R_1,...
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0answers
31 views

cycloid-based polynomials

take the set of polynomials of the form $(a+b)^n$ and generalize them: let $P_f(a,b;n)$ be a sequence of polynomials where $f:(-c,c)\to \mathbb{R}^+$ is a function with $\int_{-c}^c f=1$ and $c$ can ...
-1
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1answer
92 views

Flatness of certain quotient rings

Let $p,q \in \mathbb{C}[x,y]$ be two polynomials such that $p_xp_yq_xq_y \neq 0$ (namely, each partial derivative is non-zero). Assume that the following four conditions are satisfied: (1) $\frac{\...
3
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38 views

Martingale polynomial functions

If $B_t$ is a Brownian motion then using Hermite polynomials one can find that $$1, B_t, B_t^2-t, B_t^3 - 3tB_t,...$$ are martingales. If $X_t$ is a diffusion $dX_t = \mu(X_t,t)dt + \sigma(X_t,t)...
0
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1answer
256 views

A sufficient condition for a set of primes to be the set of reducibility of an integer polynomial

Let $P$ be the set of all positive primes. Let $S$ an arbitrary infinite subset of $P$ satisfying the following assumption: there exists a finite Galois extension $K$ of $\mathbb{Q}$ and a conjugacy ...
0
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1answer
66 views

A density zero set of primes dividing the values of a non-constant integer polynomial

For a given $P\in \mathbb{Z}[x]$ call a positive prime $p$ good if there exists $n\in \mathbb{Z}$ such that $p$ divides $P(n)$. Does there exist a non-constant $P$ such that the set of good primes has ...
3
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36 views

Bounds on degrees of minimal polynomials of infinite degree algebraic extension

If $E/F$ is algebraic extension of finite degree $n$, then if $\alpha \in E$ is an element, then the degree of minial polynomial $m_\alpha$ for $\alpha$ is at most $n$. Even better, $\deg m_\alpha$ ...
18
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2answers
1k views

Real rootedness of a polynomial

Let's consider $m$ and $n$ arbitrary positive integers, with $m\leq n$, and the polynomial given by: $$ P_{m,n}(t) := \sum_{j=0}^m \binom{m}{j}\binom{n}{j} t^j$$ I've found with Sage that for every $...
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0answers
36 views

finding a good term order for grobner basis

What are the tricks to pick a "good" monomial order to find a Grobner basis for a given ideal? By good I mean one in which the final Grobner basis has a simple expression in terms of the ...
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0answers
33 views

On a type of equations that involve certain multiplicative functions and polynomials, in relation to their number of solutions

Past weekend I was interested in the sequence A058891 from the On-Line Encyclopedia of Integer Sequences, from this, inspired by the equation due to Benoit Cloitre (2002) that shows the comments, I ...
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40 views

Coefficients of factors in terms of those of the product

I have asked this question on Math Stackexchange but received no reply, so I am trying it here. If we have two polynomials $p(x)=\sum_{i=0}^n a_i x^i$, $q(x)=\sum_{j=0}^mb_j x^j$, then the ...
2
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1answer
98 views

Classification of associative polynomial functions

What is known about a classification of associative (binary) polynomial functions? First of all, it is interesting in two cases: over Integral domain (or even over field) and over ring of integers ...
2
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0answers
71 views

Getzler's stable graphs for modular operads

In The semi-classical approximation for modular operads, Getzler displays a table at the bottom of page two enumerating certain stable graphs. (This is related to the MO-Q "Stable graphs: Feynman ...
0
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0answers
85 views

Roots of a family of 4-parameter polynomials

Let $k, \ell, p$ and $q$ be positive integers, with $q>p>1$ and $\gcd(p,q)=1$. Let $f(x)$ the polynomial given by $$ f(x)=x^q-kx^{q-p}-\ell. $$ This polynomial is related to a family of two-...
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88 views

Trasforming a system of rational equations into an equivalent system of polynomial equations

Suppose that a system of rational equations $r_1=0, r_2=0, \dots, r_m=0$ defines a zero dimensional variety $V$. Is there an algorithm to produce polynomials $p_i$, starting from the rational ...
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0answers
16 views

Best two round game organization

Consider some game (say, table tennis) with two players, in each play outcome is random and independent from previous outcomes. Assume also that probability of winning is fixed for first player and ...
9
votes
1answer
464 views

Is there a ring for which the reducibility of a polynomial is undecidable?

Let $R$ be a ring such that all of its elements have a finite number of divisors, ie $\forall r\in R\, |\{x\in R: x|r\}|<\infty$. Then we can decide whether a polynomial in $R[t]$ is reducible ...
4
votes
1answer
286 views

Questions about a return map

Consider the following map in the interval $u\in[-1,1]$ ($U\in[-1,1]$ also) $$U=u-\frac{64}{3}u^{3}+64u^{5}-64u^{7}+\frac{64}{3}u^{9}$$ It has 3 fixed points at $u=0,\pm 1$. If we compute the ...
6
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0answers
209 views

When are complex polynomial maps surjective?

Consider a complex polynomial map $f: \mathbb{C}^p \to \mathbb{C}^q$ for some $p \geq q \geq 1$ (not necessarily equal). What is a sufficient condition for $f$ to be surjective? I am aware of some ...
5
votes
1answer
191 views

Polynomial approximation (in $L^1$ norm) at minimal cost

Define the cost of a polynomial $\sum_{i=0}^N a_i x^n$ to be $\sum_{i=0}^N |a_i|$. Let $g:[0,1]\to \mathbb{R}$ be a function to be approximated — say, $g(x)=0$ if $0\leq x < e^{-1}$, $g(x)= 1/x$ ...
3
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0answers
77 views

Another application of Borel-Cantelli Lemma

I ask this question on math stackexchange, but there is no answer, so please forgive me I ask it here again. Let $c>0$ and $P(x)$ be a polynomial. Then there exists a $p>1$ (e.g. we can take $p$...
1
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0answers
53 views

How to prove the rational polynomial is nonincreasing?

Suppose $p(x)$ is a univariate real-rooted polynomial. It is easy to see that the following rational polynomial $$\Psi_p(x) = \frac{\partial^2p}{p}(x)=\sum_{1\leq j<k\leq r}\frac{2}{(z_i-\...
5
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0answers
137 views

Hahn's approach to Hilbert's 17th problem?

The Wikipedia article on Hahn Series mentions that these were studied by Hahn "in his approach to Hilbert's seventeenth problem". Is this correct? If so, what was this approach, and where can I ...
1
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1answer
38 views

Positivity in extensions of ordered fields

Let $F$ be an ordered field and $f\in F[X]$ be a polynomial such that $f(x)>0$ for all $x\in K$. Is it possible that there is an extension $L\supseteq K$ of ordered fields and $y\in L$ such that $f(...
8
votes
2answers
367 views

Supremum of $ a_n = a_{n-1}^3 - a_{n-2} $ [closed]

Let $a_1=0$ and let $ - \ln(2) < a_2 < \ln(2) $ Define $$ a_n = a_{n-1}^3 - a_{n-2} $$ Then $$ \sup_{n>2} a_n = a_2 $$ And $$ \inf_{n>2} a_n = - a_2 $$ How to prove that ?
2
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1answer
57 views

Dominant root of a family of odd degree polynomials

Let $q$ be an odd positive integer and denote by $f_q(x)=x^q-x^{q-2}-1$. This family of polynomials appears in some problems related to recurrence sequences. In order to try to use some standard ...
1
vote
2answers
106 views

Bivariate polynomial divisibility test of Spielman

Setup In his thesis (lemma 4.2.18, p. 97-98) Spielman describes a divisibility test for bivariate polynomials $E,P\in k[X,Y]$, where $k$ is a field (of positive characteristic for what I'm interested ...
6
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1answer
549 views

Fundamental Theorem of Algebra, via algebra

I know there are already a couple of questions on this on the site, but I haven't seen an answer to this particular form... We know, from the Fundamental Theorem of Algebra, that the complex ...
1
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0answers
78 views

How many residue classes mod $p$ does the image of a polynomial with integer coefficients occupy? (Status of a question of Chowla)

In his 1952 AMS Bulletin article "The Riemann zeta and allied functions" Chowla asks the following: Given a polynomial $f$ with integer coefficients, how many residue classes mod $p$ does its image ...
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0answers
39 views

Largest collection of pairwise relatively prime polynomials with bounded individual degree

Suppose $\mathcal{P}$ is a set of polynomials in $\mathbb{F}_p[x_1, \dots, x_m]$ with the properties that any $f\in \mathcal{P}$ has degree at most $d-1$ in each variable $x_i$, and any distinct $f,g\...
1
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0answers
52 views

What fraction of multivariate polynomials with bounded individual degrees are irreducible?

How many polynomials in $\mathbb{F}_p[x_1, \dots, x_m]$ with degree at most $d-1$ in each variable $x_i$ are irreducible? Here $m$ and $d$ are positive integers, $p$ is a prime, and $\mathbb{F}_p$ ...
4
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0answers
181 views

Any way around Abel's impossibility theorem?

Abel's impossibility theorem states that the roots of a general polynomial (of degree 5 or higher) cannot be written using arithmetic operations and radicals. Radicals are solutions of a specific ...
1
vote
1answer
88 views

Degree of continous function, a question about its representation

Let $f \in C(\mathbb R,\mathbb R)$, $\text{degree}(f)=\sup\limits_{a \in\mathbb R} \{ \text{card}(f^{-1}(\{a\})) \}$ Is it true that $\forall f \in C(\mathbb R,\mathbb R),\text{degree}(f)=k\in\mathbb ...
1
vote
0answers
130 views

Chebyshev polynomials from roots [closed]

Given a polynomial $P_n(x)$ with roots at $\{a_i\}$: $P_n(x)=\prod\limits_{i=1}^{i=n} (x-a_i)$ one can obtain the Chebyshev coefficients from the roots. It is implemented eg in python. Can we ...
0
votes
1answer
97 views

Diophantine equations that involve cubes and the volume of square frustums

This week I wondered about diophantine problems that involve the volume of certain cubes and frustums, see the Wikipedia Frustum. I wondered if each one of these problems have infinitely many ...
4
votes
0answers
128 views

Finding roots of this polynomial

I have a polynomial, given for parameters $x$ in $\mathbb{R}_+$ and $\alpha$ and $\beta$ in $\mathbb{R}_+^{n}$ by : $$P(t) = \sum\limits_{i=1}^n \left\{\left(\beta_i - \frac{\alpha_i}{x} - t\right) ...
1
vote
1answer
113 views

Generate a two-variable polynomial from its "roots [closed]

I know, from mathematics basis, that a polynomial with one variable can be factor in function of its roots, so I can generate any one-variable polynomial from its zeros. But I want know if is ...
0
votes
1answer
60 views

Existence of element $(x_0,y)$ in a set of common zeros for all $(x_0,y)$ satisfying system of inequalities

Let $f_1,f_2,\cdots,f_n,g_1,g_2\cdots,g_m\in \mathbb{R}[x,y]$, then define the affine variety and semi-affine variety as follows: $V(f_1,f_2,\cdots,f_n):=\{(x,y)\in\mathbb{C}^2: f_1(x,y)=f_2(x,y)=\...
0
votes
1answer
110 views

Special elements of the Cremona group

After asking this MO question, I wish to ask about the following special case: Let $f$ be a $\mathbb{C}$-algebra automorphism of $\mathbb{C}(x,y)$ and denote $u:=f(x),v:=f(y)$. Is it possible to ...
3
votes
1answer
238 views

“Non-associative” standard polynomials

I saw somewhere (I appreciate if anyone has any references to proof of this fact) that if $A$ is a finite dimensional associative algebra such that $\textrm{dim}(A)<n$, then $A$ satisfies the ...
11
votes
3answers
1k views

How can I simplify this sum any further?

Recently I was playing around with some numbers and I stumbled across the following formal power series: $$\sum_{k=0}^\infty\frac{x^{ak}}{(ak)!}\biggl(\sum_{l=0}^k\binom{ak}{al}\biggr)$$ I was able ...
2
votes
0answers
75 views

Saturation with respect to $f$ of an ideal $I$

According to wikipedia, the saturation with respect to $f$ of an ideal $I$ in $R$ is the ideal $I:f^\infty:=\{g\in R:\exists k\in\mathbb{N}, f^kg\in I\}$. The important property of the saturation, ...
1
vote
1answer
188 views

Integral zeros of a multivariate polynomial

Consider the multivariate polynomial $$f(x_1,\ldots,x_m)=mk\sum_{i=1}^mx_i^2-mk(k-1)\sum_{i=1}^mx_i-\left(\sum_{i=1}^mx_i\right)^2,$$ for integers $m,k\ge2$. We are looking for integral zeros of $f$ ...
9
votes
2answers
259 views

Can we recover all $k$-minors of a square matrix from some of them?

This is a cross-post. Let $k,n$ be natural numbers, $1<k<n$. Suppose we have an "unknown" invertible $n \times n$ matrix $A$ over a field of characteristic zero. (we do not know the entries of ...

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