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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Irreducibility of q-factorial plus 1

Let $q$ be a formal variable and for every positive integer $n$ let $$[n]_q! = 1 (1 + q)(1 + q + q^2) \dotsm (1 + q + \dotsb + q^{n-1})$$ be the $q$-factorial. Is it true that $[n]_q! + 1$ is an ...
Penchez's user avatar
  • 341
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0 answers
699 views

Solving polynomial systems with homotopy. Where is the bottleneck?

I have a polynomial system with $n+k$ unknowns ($n+k$ can be greater than 8), that is known to have a limited number of isolated solutions. I want to solve this system numerically, but if I plug it ...
Ulderique Demoitre's user avatar
14 votes
3 answers
9k views

Can you efficiently solve a system of quadratic multivariate polynomials?

Given a system of 2nd-degree polynomials, $P=\{p_1,\dots,p_m\}$ where $p_i: \mathbb{R}^n \rightarrow \mathbb{R}$, can you efficiently find a common zero of all of these polynomials? In other words, ...
Arc's user avatar
  • 243
14 votes
3 answers
3k views

Truncated Exponential Series Modulo $p$: Deeper meaning for a Putnam Question.

Apparently B6 of the Putnam this year asked: Suppose $p$ is an odd prime. Prove that for $n\in \{0,1,2...p-1\}$, at least $\frac{p+1}{2}$ of the numbers $\sum^{p-1}_{k=0} k! n^{k}$ are not divisble ...
Eric Naslund's user avatar
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14 votes
5 answers
971 views

How can I write down polynomial relations that define when a polynomial is a square?

It's easy to tell when a polynomial is squarefree (or not): that's just the question of the vanishing of the discriminant, which can be dealt with as the resultant of $f$ and $f'$. However, given a ...
Charles Siegel's user avatar
14 votes
2 answers
1k views

Minimal polynomial with a given maximum in the unit interval

Find the lowest degree polynomial that satisfies the following constraints: i) $F(0)=0$ ii) $F(1)=0$ iii)The maximum of $F$ on the interval $(0,1)$ occurs at point $c$ iv) $F(x)$ is positive ...
David Shor's user avatar
14 votes
6 answers
3k views

On Polynomials dividing Exponentials

EDIT, May 2015: in the second edition of the relevant book, the question was corrected, a single number had been mis-typed. The corrected question (thanks to Max) is to find all positive integer pairs ...
Daniel Kohen's user avatar
14 votes
3 answers
2k views

When does $P(a−b)=0$ for $a≠b$ ensure $P(0)=0$? (Continued.)

As a natural (and expectable) extension of my earlier question: How large must be a set $A\subset F_2^n$ to ensure that if $P$ is a cubic polynomial in $n$ variables over the field $F_2$, vanishing ...
Seva's user avatar
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14 votes
2 answers
2k views

Polynomial values are powers of two

The initial question comes from Komal in 1999. Namely it asks to show that for infinitely many $n$ there is a polynomial $f\in\mathbb{Q}[X]$ of degree $n$ such that $f(0),f(1),\dotsc,f(n+1)$ are ...
Vlad Matei's user avatar
14 votes
1 answer
2k views

Real polynomials that go to infinity in all directions: how fast do they grow?

Let $f(x_1, \cdots, x_n) \in \mathbb{R}[x_1, \cdots, x_n]$ be a polynomial. Define property $\mathbf{P}$ to be the property that there exists a compact set $K \subset \mathbb{R}^n$ and a positive ...
Stanley Yao Xiao's user avatar
14 votes
4 answers
2k views

Is a polynomial with 1 very large coefficient irreducible?

I am asking for some sort of generalization to Perron's criterion which is not dependent on the index of the "large" coefficient. (the criterion says that for a polynomial $x^n+\sum_{k=0}^{n-1} a_kx^k\...
Gjergji Zaimi's user avatar
14 votes
3 answers
2k views

Polynomials that are sums of squares

Is any algorithm known for determining whether or not a multivariate polynomial with integer coefficients can be written as a sum of squares of such polynomials? By way of background, if we one ...
raffer's user avatar
  • 141
14 votes
1 answer
961 views

Positive roots of a polynomial

Let $a_i>0$, $i=1,\dots,n$, and put $\overline{a}:=\frac{1}{n}\sum_{i=1}^n a_i$. Assuming not all $a_i$'s are equal, take $$ p(x):=\sum_{i=1}^n a_i (a_i-\overline{a})\prod_{k=1,\dots,n\;k\neq i} (x+...
dima's user avatar
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14 votes
3 answers
1k views

Probability of coprime polynomials

Given positive integer $N$, we choose $m_1, m_2, n_1, n_2$ independently and with equal probabilities from $\{0,1,\ldots,N\}$, and let $f_1 = x^{m_1} + (1+x)^{n_1}$ and $f_2 = x^{m_2} + (1+x)^{n_2}$ ...
Robert Israel's user avatar
14 votes
3 answers
907 views

Sets that can be mapped onto R^n by a polynomial

The question was edited several times. Most recent version, suggested by Fedja: Does there exist an open set $U\subset \mathbb R^n$ (n>1) that contains balls ...
14 votes
2 answers
1k views

Number of nonzero terms in polynomial expansion (lower bounds)

Let $f(x) = a_1x^{z_1} + a_2x^{z_2} + \cdots + a_kx^{z_k}$ be a polynomial with coefficients $(a_1, \ldots, a_k) \in \mathbb{F}_q^*$ and $z_i$ are distinct positive integers. If I need to compute the ...
Lucas Perin's user avatar
14 votes
1 answer
1k views

Do almost all systems of quadratic equations have solutions?

If I have a system of linear equations, $A x = c$, with $A$ an $n\times n$ complex matrix, it is relatively easy to see that the set of matrices $A$ for which there is no (complex) solution has ...
glS's user avatar
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14 votes
1 answer
463 views

Polynomials for which $f''$ divides $f$

Let $n \geq 2$ and let $a < b$ be real numbers. Then it is easy to see that there is a unique up to scale polynomial $f(x)$ of degree $n$ such that $$f(x) = \frac{(x-a)(x-b)}{n(n-1)} f''(x).$$ ...
David E Speyer's user avatar
14 votes
1 answer
497 views

Is a polynomial group law on $\mathbb{R}^n$ automatically nilpotent?

I was told that a polynomial group law on (all of) $\mathbb{R}^n$ gives automatically a nilpotent (Lie, of course) group. Is it true? Where can I find a proof? A counterexample for open subsets of $...
Gian Maria Dall'Ara's user avatar
14 votes
2 answers
571 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 $n$...
MannyC's user avatar
  • 243
14 votes
4 answers
2k views

Two questions about discriminants of polynomials in ℚ[x]

Suppose $f \in \mathbb{Q}[x]$ is monic, with roots $\alpha_1,\dotsc,\alpha_n$. Define the discriminant of $f$ to be the number $ \Delta = \prod_{i<j} (\alpha_i - \alpha_j)^2$. Let $D(f) = \sqrt{\...
Lee's user avatar
  • 141
14 votes
1 answer
427 views

Converse of the Lee-Yang circle theorem for polynomials with unitary roots

The Lee-Yang circle theorem states that if $\left( a_{ij} \right)$ is a Hermitian square $n \times n$ matrix whose entries are in the closed unit disc, then the polynomial $$ P\left(Z \right) = \sum_{...
darkl's user avatar
  • 680
14 votes
4 answers
2k views

Computing minimal polynomials using LLL

I would like to use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL algorithm) to compute the minimal polynomial of a (real) algebraic number $\alpha$ from a decimal approximation $a$...
Mark Bell's user avatar
  • 3,125
14 votes
2 answers
532 views

regular polygon question

Let $a_1,a_2,\ldots,a_n$ be distinct points on the complex plane $\mathbb{C}$ and $L$ be a circle in $\mathbb{C}$ such that $$f(z):=\sum_{i=1}^n|z-a_i|^{2n-2}$$ is constant on $L.$ Could somebody ...
Ma Na's user avatar
  • 309
14 votes
1 answer
765 views

Theorems proved using combinatorial nullstellensatz that have no other known proof

Alon's (or Alon and Tarsi's?) combinatorial nullstellensatz is a powerful algebraic tool with many applications in combinatorics and number theory. See this, this, this and this mathoverflow question. ...
Anurag's user avatar
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14 votes
1 answer
500 views

Hyperbolic polynomials and group representations

Recall that a homogeneous polynomial $P\in{\mathbb R}[X_1,\ldots,X_d]$ of degree $n$ is hyperbolic in the direction of a vector $V\ne0$ if for every vector $W$, the univariate polynomial $t\mapsto P(W+...
Denis Serre's user avatar
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14 votes
1 answer
2k views

Elementary proof for Hilbert's irreducibility theorem

I have tried to find a complete proof for Hilbert's irreducibility theorem, but everything I found was way beyond my level of understanding. I am only interested in the simple case where the ...
kiena's user avatar
  • 141
14 votes
2 answers
1k views

Symmetric group action on squarefree polynomials

The following dynamical system on polynomials comes mostly from idle curiosity, but I hope it is of some interest. Background Fix some natural number $n$. Let $P$ be the quotient of the polynomial ...
Vidit Nanda's user avatar
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13 votes
2 answers
1k views

Is irreducibility of polynomials $\in \mathbb{Z} [X]$ over $\mathbb{Q}$ an undecidable problem?

There are a number of criteria for determining whether a polynomial $\in \mathbb{Z} [X]$ is irreducible over $\mathbb{Q}$ (the traditional ones being Eisenstein criterion and irreducibility over a ...
SARTHAK GUPTA's user avatar
13 votes
5 answers
3k views

A geometric proof of the Gauss-Lucas theorem

Motivated by a geometric proof of the Fundamental Theorem of Algebra we ask: Is there a geometric proof for the Gauss-Lucas theorem? Since we are working on a half plane, can one imagine a possible ...
Ali Taghavi's user avatar
13 votes
2 answers
2k views

About irreducible trinomials

This question was inspired by this one. For every $n>m>0$ consider the polynomial $p_{m,n}=x^n-x^m-1$. For which $m,n$ is $p_{m,n}$ irreducible over $\mathbb Q$? In particular, if $m$ is odd,...
user avatar
13 votes
2 answers
2k views

Irreducible polynomial over number field with roots in every completion?

Let K/Q be a field, probably not a finite extension. Is it possible for a polynomial to be irreducible over K but have a root in every completion of K? What about all but finitely many completions? ...
Dror Speiser's user avatar
  • 4,563
13 votes
5 answers
3k views

Application of polynomials with non-negative coefficients

Question 1: Are there any deeper applications (in any field of mathematics) of polynomials (with possibly more than one variable) over the real numbers whose coefficients are non-negative? So far I ...
Miroslav Korbelar's user avatar
13 votes
5 answers
5k views

Number of spanning forests in a graph

Hello, I have two questions that have been bugging me recently. The first is about the number of spanning forests in a graph and the second is about enumerating these with edge labels. Q1: I am ...
Aleks Vlasev's user avatar
13 votes
3 answers
1k views

Height of cyclotomic polynomials

Recall that the cyclotomic polynomial of order $n$ is $$\Phi_n(X)=\prod_{gcd(k,n)=1}(X-e^{2ik\pi/n}).$$ Its degree is $\phi(n)$ (Euler's indicator). Inversion of $$X^n-1=\prod_{d|n}\Phi_d(X)$$ by the ...
Denis Serre's user avatar
  • 51.5k
13 votes
3 answers
444 views

Proportion of polynomials of a fixed degree with a certain number of real roots

For a polynomial $f(x) = \sum_{i=0}^dc_ix^i \in \mathbb Z[x]$ of degree $d$, let $$ H(f):=\max\limits_{i=0,1,\ldots, d}\{|c_i|\} $$ denote the naive height. Further, define $$ R(M, r, d) := \#\{f(x)...
Anton's user avatar
  • 1,573
13 votes
3 answers
704 views

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

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 ?
mick's user avatar
  • 703
13 votes
1 answer
776 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 ...
Lucio Tanzini's user avatar
13 votes
4 answers
1k views

Showing that a family of polynomials has positive and real roots.

Hi everybody, for my research I am dealing with the following function: $$\alpha_n(x):=\left.\frac{\partial^{2n+1}}{\partial z^{2n+1}}\frac{\sinh(z)}{\cosh(z)-1+x}\right|_{z=0},\quad n\in \mathbb{N},...
Enzo's user avatar
  • 131
13 votes
1 answer
3k views

When are complex polynomial maps almost surjective?

Consider a complex polynomial map $f: \mathbb{C}^n \rightarrow \mathbb{C}^n$. For $n = 1$, the fundamental theorem of algebra says that, for any $y \in \mathbb{C}$ there exists $x \in \mathbb{C}$ ...
sreekanth's user avatar
  • 133
13 votes
1 answer
980 views

Is -1 a sum of 2 squares in a certain field K?

Consider the field of fractions $K$ of the quotient algebra $\mathbb{R}[x,y,z,t]/(x^2+y^2+z^2+t^2+1)$, where $\mathbb{R}$ is the field of real numbers and $x,y,z,t$ are variables. Clearly $-1$ is a ...
Mikhail Borovoi's user avatar
13 votes
2 answers
1k views

Families of quintics in $\mathbb{Q}[x]$ with Galois group $A_5$

Theorem. The Galois group of a quintic polynomial $f\in\mathbb{Q}[x]$ is $A_5$ if and only if its discriminant is a rational square and its Weber sextic resolvent has no rational root. Question. What ...
Favst's user avatar
  • 1,985
13 votes
3 answers
834 views

Effective algorithm to test positivity

Let $f(x_1,\ldots, x_n)$ be a real polynomial in several variables. Is there an effective algorithm to test whether $f$ is positive (or nonnegative) on the whole of ${\mathbb{R}}^n$?
user avatar
13 votes
2 answers
495 views

Roots of lacunary polynomials over a finite field

If $P$ is a polynomial over the field $\mathbb F_q$ of degree at most $q-2$ with $k$ nonzero coefficients, then $P$ has at most $(1-1/k)(q-1)$ distinct nonzero roots. Does this fact have any standard ...
Seva's user avatar
  • 22.8k
13 votes
2 answers
537 views

$f$ real-rooted forbid truncated $\frac1f$ to be so?

Let $f(x)$ be a polynomial in the ring $\mathbb{R}[x]$, the roots are all real and $f(0)=1$. Write the Taylor series of $1/f(x)$ around the origin as $$\frac1{f(x)}=\sum_{k=0}^{\infty}a_kx^k,$$ and ...
T. Amdeberhan's user avatar
13 votes
3 answers
2k views

Which polynomials are determinants of a symmetric matrix with linear entries?

Let $k$ be a field. Can each degree $n$ polynomial $P(t) \in k[t]$ be written as the determinant of the matrix $A + tB$, where $A$ and $B$ are two symmetric $(n \times n)$-matrices with entries in $k$?...
Wanderer's user avatar
  • 5,113
13 votes
3 answers
3k views

Density of Irreducible Polynomials in $\mathbb{Z}[x]$

Recently I was thinking about some questions concerning $\mathbb{Z}[x]$ and realized that they might be a bit easier if I knew the relative densities of reducible polynomials. Let $P_d$ denote the ...
ARupinski's user avatar
  • 5,181
13 votes
3 answers
914 views

Polynomials vanishing modulo some integer $n$

It is well-known that a polynomial $q \in \mathbb Z[t]$ vanishes modulo $p$ only if it lies in the ideal $J_p$ generated by $p$ and $t^p-t$. This means that either the degree is large (at least $p$) ...
Andreas Thom's user avatar
  • 25.3k
13 votes
1 answer
1k views

Irreducibility of Schur polynomials

A natural question covering both this and this question would be Let $n>2$. Describe Young diagrams $\lambda$ with at most $n$ nonempty rows (or equivalently non-increasing sequences $\lambda=(\...
Vladimir Dotsenko's user avatar
13 votes
3 answers
1k views

When are Ehrhart functions of compact convex sets polynomials?

Given a lattice $L$ and a subset $P\subset \mathbb R^d$, we define for each positive integer $t$ $$f_P(L,t)=|(tP\cap L)|$$ the number of lattice points in $tP$. Let's say $P$ is nice if $f_P(L,t)$ is ...
Gjergji Zaimi's user avatar

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