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.
2,555
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A problem on polynomials
Let $P(z)$ be a polynomial of degree $n$ with $|P(z)|\leq 1$ on $|z|=1$ and $P_m(z)$ be a partial sum of $P(z).$ How large $P_m(z)$ can be on $|z|=1?$
- 559
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1
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What does it mean polynomials share Newton polytope?
I have trouble understanding the connection between polynomials and Newton polytopes. I will try to make a short introduction to my problem and hope you will catch on. In the end I will ask questions.
...
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Irreducible Hurwitz Factorization of A Complex Polynomial
I've decided to repost this question, which originally appeared on MSE, here. It is part of my series of open problems for enthusiasts and, while I understand this crowd is focused on professionals, ...
- 263
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1
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454
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Stronger versions of Schwartz-Zippel for random linear subspaces
This is a (self-contained) followup question to https://math.stackexchange.com/questions/380672/analogue-of-the-schwartz-zippel-lemma-for-subspaces.
Let $f : \mathbb{R}^n \to \mathbb{R}$ be a nonzero ...
- 1,758
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3
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Are there any neat algorithm to factor a homogenous polynomial, given we know this polynomial factors into linear forms?
Are there any neat algorithm to factor a multivariable (more than 2 variables) homogenous polynomial, given we know this polynomial factors into linear forms?
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A known Lemma on the largest root of a polynomial and its derivatives?
Greetings,
I am currently working on a paper that involves an upper bound of the largest root of a polynomial. With the help of the Mean Value Theorem, I believe a colleague and I have proved the ...
- 43
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4
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Systems of polynomial equations
Hi all,
I'm an engineer assigned to determine some parameters of a manipulator (ie., calibration). It has a number of parameters, but after some manipulations of its dynamic equations, I can have the ...
- 41
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116
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How are Lie groups and polynomial resolvents related?
I came across the following sentence in Stevenhagen and Lenstra's wonderful little article Chebotarëv and his density theorem:
Nikolai's interest in [polynomial] resolvents led him to study Lie ...
- 173
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1
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240
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Irreducible integral polynomials having roots module primes in arithmetic progressions
Let $f(x)$ be an irreducible polynomial with integer coefficients. One can show (see Exercise 7.2 in this paper of Lenstra) that if $f(x)=0$ has a solution mod $p$ for all but finitely many primes $p$...
- 6,104
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Software computing dimension and degree
Assume a projective scheme $X_{k_1,\dots,k_r}\subset\mathbb{P}^n$ is given as the set of common solutions of homogeneous polynomials $F_1(x_0,\dots,x_n),\dots,F_s(x_0,\dots,x_n)$, where the $F_i$ ...
user114666
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450
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All rational periodic points
I am trying to find all rational periodic points of a polynomial. To specify: a periodic point is the point that satisfy $f^n(x)=x$. It is related to dynamical systems in fact. So the current codes ...
- 41
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Jacobian criterion for algebraic independence over a perfect field in positive characteristics
It is well known that the Jacobian criterion for algebraic independence does not hold in general for fields of positive characteristics. However, the following partial statement seems promising:
...
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374
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A possible generalization of Gauss Lucas theorem to higher dimension
A real half space in $\mathbb{C}^2$ is $$\{(z,w)\in \mathbb{C}^2\mid \phi(z,w) > \lambda\}$$ where $\lambda$ is a real number and $\phi$ is a $\mathbb{R}$- linear functional from $\mathbb{C}^2$ to $...
- 312
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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 ...
- 213
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407
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Cipolla's Prime numbers function: Computing the coefficients of the polynomial
In many publications dealing with asymptotic determinations of the n-th prime number, the following paper is cited:
[1]: M. Cipolla, La determinazione asintotica dell’n-esimo numero primo. rendiconti ...
- 143
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2
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351
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Linear operator on polynomials and invariant sets of roots
Let $T:\mathbb{C}_n[x] \to \mathbb{C}_n[x]$
be a linear map from the vector space of polynomials of degree $n$ to itself.
Let $S \subset \mathbb{C}$ be a set with at least $3$ points, such that
for ...
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4
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2
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123
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Univariate polynomial interpolation with restricted degrees
Let $D=\{d_1, d_2, \ldots, d_n\}$ be an integer set. I'd like to know if I can interpolate any collection of $n$ points $(x_1, y_1), (x_2, y_2), \ldots, (x_{n}, y_{n})$ by a polynomial whose degree ...
- 41
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Two bivariate polynomials (or rational functions) that generate $\mathbb{C}(x,y)$
Let $f=f(x,y),g=g(x,y) \in \mathbb{C}[x,y]$, each of degree $\geq 1$, and $f,g$ are algebraically independent over $\mathbb{C}$ (= their Jacobian $\in \mathbb{C}[x,y]-\{0\}$).
(1) Is there a ...
- 2,783
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Semialgebraic sets containing irrational power functions
Let $\alpha$ be an irrational number, and consider the set $A=\{(x,x^\alpha),x\ge 0\}\subseteq \mathbb{R}^2$, which is the graph of the function $f(x)=x^\alpha$.
I'm trying to prove/disprove the ...
- 203
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303
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Degrees of polynomials vanishing to various orders on a set of points
Suppose $X$ is a finite set of points in $\mathbb C^n$. Let $d_r$ denote the minimum degree of a polynomial vanishing to order $r$ at each point of $X$. By linear algebra, we know find can find a ...
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If $\{f\in R[x]\:|\:f\text{ monic}\}$ is a right denominator set, is $\{f^i\:|\:i\geq 0\}$ a right denominator set also?
Let $R$ be a right (and left) Noetherian ring and $T=R[x]$ its polynomial ring. It was shown by Stafford that the set $S=\{f\in T\:|\:f\text{ monic}\}$ is a right denominator set. So my question is, ...
- 348
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227
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Product of polynomial coefficients of a recurrence
A recurrence is given by
$f[0]=2x$, $f[1]=3x^3-x^2+x+1$,
$$
f[n]=(x^{2^n}+1)f[n-1]+(x^{2^n}+1)(x^{2^n-1}+1)
$$
How does the PRODUCT of the nonzero coefficients of $f[n]$ scale with $n$?
4
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Zero measurability of zero-sets of polynomials
Could anyone point me to a reference showing that the zero set of a polynomial in $n \ge 2$ variables has Lebesgue measure zero? I wonder if there are pathological examples, and some conditions ...
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2
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505
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Closed-Form solution for system of simple nonlinear equations
I am interested in analytical solutions for a system of nonlinear equations.
(The question was first asked at math.SE, where (after 1months and one rounds of bounty) there is only interesting ...
- 155
4
votes
1
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310
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Irreducible monic polynomials
I am looking for criteria for the irreducibility of monic polynomials with constant term $\pm1$ over $\mathbb Q$. Eisenstein's criterion clearly doesn't apply here.
For instance, for the family of ...
- 2,085
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1k
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Existence of real solutions for a system of linear and quadratic equations
Suppose we have a system of linear and quadratic equations with rational coefficients and we want to find out whether this system has a real solution or not. The actual values of a solution is not ...
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3
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708
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The Poisson-kernel in the plane and polynomials
Let
\begin{align*}
p(z) & = \ \prod_{j = 1}^{n} (z - z_j) \, ; \ |z_j| \ = \ 1 \\
& = \ \prod_{j = 1}^{l+1} (z - z_j)^{M_j}
\end{align*}
be a non-constant complex polynomial with $l+1$ ...
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162
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If the sequence of degrees of the iterates of a self-map of $\mathbb{A}^2$ is bounded, is it eventually periodic?
Let $f : \mathbb{A}_k^2 \to \mathbb{A}_k^2$ be a regular self-map of the affine plane over a field $k$ of characteristic zero. Assume that the sequence $(\deg{f^n})_{n \in \mathbb{N}}$ is bounded. Is ...
- 13.7k
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Criteria for system of parameters in polynomial rings
Let $k$ be a field and let $R=k[x_1,...,x_n]$ be a polynomial ring over $k$. A subset $\lbrace y_1,...,y_n\rbrace$ of $R$ is called a homogenous system of parameters (hsop) of $R$, if
the $y_i$ are ...
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Irreducibility of some trinomials modulo $p$
Let $n>1$ be an integer. An old result of Selmer,
See Theorem 1, page 289 in
http://www.mscand.dk/article.php?id=1472,
(If the link does not work try googling: ...
- 1,601
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Divisibility relation with a specific sum of divisors
Let $\sigma(n)$ be the sum of the divisors of $n$. Is it always true that if $n$ is odd, that $$n\mid\sum_{k=1}^{\frac{n-1}{2}}k^2\sigma(k)\sigma(n-k)?$$
I have checked this up to $n=100$, and I ...
- 6,100
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159
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Any ideal as an intersection of ideals primary to maximal ones
The Nullstellensatz says that any ideal $I \subset \mathbb{C}[x_1,\dots,x_n]$ has the property that
$\sqrt{I} = \bigcap_{\text{maximal } \mathfrak{m} \supset I} \mathfrak{m}$
Is it also true that we ...
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Why do these polynomials split almost in the middle?
Start with a palindromic sequence of integers $(a_0, a_1, \ldots, a_{n+1})$, i.e. $a_j=a_{n+1-j}$, and put $a_j:=0$ for $j<0$ and $j>n+1$. You may readily guess that the choice of the binomial ...
- 13.2k
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1
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Distance to finite degree polynomials for BV functions
A result of Jackson establishes for lipschitz functions $f\in\text{W}^{1,\infty}(0,1)$ the bound $$\inf_{p\in\mathbf{R}_n[x]} \|f-p\|_\infty\lesssim \frac{1}{n}\|f'\|_\infty,$$
where $\mathbf{R}_n[x]$ ...
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1
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250
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Weierstrass approximation theorem proof by Gram-Schmidt orthogonal polynomials?
I am familiar with Bernstein proof of Weierstrass Theorem. However, I am curious whether it can be proven by using Gram-Schmidt orthogonalization of $x^n, n=0,\dots,N$ monomials? As $N\rightarrow\...
4
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1
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Why does this "factorial sequence" appear in the OEIS?
For a reciprocal of a polynomial, $f = \frac{1}{p}$, we (presumably) may construct a sequence $(c_n)_{n=0}^\infty$ such that for all $N\ge 0$
$$f(k)k! = \sum_{n=0}^{N-1} c_n(k-n)! + O((k-N)!). $$
I ...
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1
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100
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Literature on the polynomials and equations, in structures with zero-divisors
I need literature about zeroes of polynomials and equation resolution in associative algebraic structures with zero-divisors, but I am having difficulties to find it.
For example, there is literature ...
4
votes
1
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239
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Is there any Menelaus-type theorem for polynomials?
Consider $n+3$ polynomials of degree $n$, say $P_1(x) , ... P_{n+3}(x)$.
In addition, consider that there are distinct (it is not necessary that all of them be distinct) numbers $x_{ij}$ for $1 \...
- 550
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1
answer
281
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$L^1$ norm of Littlewood polynomials on the unit circle
A Littlewood polynomial is a polynomial with coefficients from $\{ 1, -1\}$ and the set of Littlewood polynomials with degree $n$ is denoted by $\cal{L}_n$.
I'm interested in a "good" lower bound on ...
- 4,454
4
votes
1
answer
410
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Cancellation problem for Laurent polynomial rings and power series rings
Throughout, let $k$ be an algebraically closed field. For two $k$-algebras $A,B$ let us write $A \cong_k B$ to mean that $A,B$ are isomorphic as $k$-algebras.
It is known that if $A$ is an integral ...
user111524
4
votes
2
answers
332
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algorithm for finding radical expressions of all conjugates of an arbitrary algebraic number expressed in radicals
By an algebraic number expressed in radicals, I mean one that is an element of a set $S$ characterized as follows:
$\mathbb{Z}\subset S$.
For any $a,b\in S$, $a+b,a·b\in S$.
For $a,b\in S$ with $b\...
- 151
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1
answer
207
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Proximity of solutions to system of degree two polynomials
Everything in this post is over the complex numbers. I would like to know if for every $\epsilon > 0$ there exists $\delta > 0$ such that the following holds for every $n$ and every $d$ which is ...
- 1,689
4
votes
1
answer
112
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Bounding the value of a polynomial map by the distance from the variety
Let $f_1,\ldots,f_r\in\Bbb C[x_1,\ldots,x_n]$ define a map $f\colon\Bbb C^n\to\Bbb C^r$. Let $Z:=Z(f_1,\ldots,f_r)\subseteq \Bbb C^n$ be the variety cut out by the $f_i$, and assume that $Z$ is ...
- 4,822
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1
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372
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Counting couples of square-free polynomials over finite fields
I have a curve defined by the following equations over the finite field $\mathbb{F}_q$ with $q=p^r$ with $p \geq 3$:
$$C_{h_1,h_2}:\begin{cases} y_1^2=h_1(t) & \\
y_2^2=h_2(t) &...
- 41
4
votes
1
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411
views
q-th powers and roots of polynomials
Let $p,q,r$ be integers with $r\ge2$; let $f$ be a polynomial of the form $f(X) = g((X+1)^r)$, which is not a $q$-th power. Let $\omega$ be a $p$-th root of unity.
Prove or disprove that the ...
- 2,489
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votes
1
answer
408
views
Convergence of the Double Integral of a Polynomial Reciprocal
Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions:
(i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$;
(ii) $f$ is non-degenerate, in the sense that there isn't a ...
- 3,132
4
votes
1
answer
691
views
Cyclotomic polynomials coprime to a fixed polynomial
Let $f \in \mathbb{Z}[x]$ be monic, irreducible and hyperbolic (no roots of absolute value $1$), and such that $f(0)= \pm 1$.
Denoting as $c_{p}(x)$ the cyclotomic polynomial $$c_{p}(x)=1+x+\cdots +x^{...
4
votes
1
answer
462
views
Cubic curve closest to the given set of points
Assume we are given the set $S$ of $n$ points on the real plane and want to draw a parametrized cubic curve (actually a segment of Bézier spline) with fixed startpoint in such a way, that the ...
- 41
4
votes
1
answer
584
views
Explicit and fast error bounds for approximating continuous functions
Main Question
This question is about finding explicit, calculable, and fast error bounds (no hidden constants) when approximating continuous functions with polynomials or simpler functions to a user-...
- 637
4
votes
1
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269
views
The highest power of $2$ dividing a polynomial evaluated at $x=3$
Let $\nu_2(a)$ be the $2$-adic valuation of an integer $x$, i.e. the largest power $t$ such that $2^t$ divides $x$.
Define the operator $D=x\frac{d}{dx}$ and the polynomial $\Phi_k(x)=\frac{x^{k+1}-1}{...
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