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
questions
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84
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Polynomials of integer coefficients that evaluated at Mersenne or Fermat numbers produce square-free integers
Mersenne numbers $M_n=2^n-1$ and Fermat numbers $F_n=2^{2^n}+1$ draw the attention of professional mathematicians to get prime constellations or statements related to primality tests for these ...
5
votes
0
answers
162
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Example of applying real quantifier elimination algorithm for polynomials
Sorry if any of this is unclear, or doesn't make much sense, I'm still trying to figure it out, a practical example such as this would likely help me understand better than anything else. I have read ...
- 51
5
votes
2
answers
353
views
Existence of algebraic integers with certain properties
Is the following statement true?
($\star$) Given integers $n > k > 0$, there exists a monic polynomial of degree $n$ with integer coefficients and constant term $\pm 1$, irreducible over $\...
- 2,411
3
votes
0
answers
89
views
The rank of a special matrix
Suppose that $P$ is a polynomial of degree $d:=\deg P$ over a field $\mathbb F$ of zero characteristic, splitting completely into pairwise distinct linear factors, and $B,C\subset\mathbb F$ are sets ...
- 22.8k
5
votes
1
answer
439
views
Abel-Ruffini theorem for systems of polynomial equations
I know the Abel-Ruffini theorem, which states that a general polynomial equation in one variable with degree $\geq 5$ is not solvable in radicals. I wonder whether there is a similar result for ...
0
votes
1
answer
246
views
If the coefficient of the polynomial positive
I want to know what is following sum coefficient looks like. We sum over all integers $p$, $q$ also we put the condition that $q$ is even. Also, it should depend on the parity of $k$
$$\bar{S}(k)=\...
- 685
2
votes
0
answers
52
views
The graph polynomial of the Total Graph of a Graph
Consider the Total Graph ($T(G)$) of a graph $G$ with vertex set $V$ edge set $E=\{(u,v)\}$ with Line graph $L(G)$ and subdivision graph $S(G)$ (formed by putting a vertex in each edge of the original ...
- 2,027
4
votes
0
answers
212
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Divisibility of the derivatives of a polynomial
(Related to MO Problem 9924.)
Suppose that $f$ is a polynomial of degree $d>0$ over a field of zero characteristic.
It is not difficult to see that if $Q$ is yet another, non-constant polynomial, ...
- 22.8k
2
votes
0
answers
41
views
Concerning $(x,y) \mapsto (x^{\frac{n}{r}+1}y + A,\mu x^{-\frac{n}{r}}+B)$
Let $r \in \mathbb{N}-\{0\}$.
Commutative case:
Let $f : (x,y) \mapsto (p,q)$ be a map from $\mathbb{C}[x,y]$ to $\mathbb{C}[x^{1/r},x^{-1/r},y]$ satisfying the following two conditions:
(i) $\...
- 2,783
-1
votes
1
answer
134
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Integral zeros of the Newton polynomial
I'm trying to understand the following result;
Statement: A newton polynomial of the form
$$a_1 {x\choose c_1}+a_2{x\choose c_2}+a_3{x\choose c_3}+⋯+a_s{x\choose c_s},$$
where $0 ≤c_1<c_2<c_3&...
3
votes
1
answer
245
views
Сomplement of the set of numbers of the form $ 4mn - m - n$?
Numbers of the form $4mn-m-n$ where $m,n\in\mathbb{Z}^+$ are
$$
A=\{2, 5, 8, 11, 12, 14, 17, 19, 20, 23, 26, 29, 30, 32, 33, 35, \ldots\}
$$
The set complement of the above set is
$$
B=\{1, 3, 4, 6, ...
- 841
9
votes
2
answers
954
views
Polynomials that share at least one root
This is a generalization of an MSE question,
Polynomials that share at least one root.
Let $P(x)$ be a specific polynomial of degree $d$, with given
real coefficients $A_i$ ($A_d=1$), and real roots:
...
- 149k
5
votes
0
answers
193
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A non-commutative analog of a result concerning a Jacobian pair
Let $k$ be a field of characteristic zero and let $E=E(x,y) \in k[x,y]$.
Define $t_x(E)$ to be the maximum among $0$ and the $x$-degree of $E(x,0)$.
Similarly, define $t_y(E)$ to be the maximum among $...
- 2,783
3
votes
0
answers
241
views
Interlacing sequences by polynomials?
Given $t=2^\ell$ where $\ell\in\mathbb N_{>0}$ and $M\in\mathbb Z$ and two sets of integers $\{a_1,\dots,a_t\}$ and $\{b_1,\dots,b_t\}$ with $0<a_1\leq \dots\leq a_t<M$ and $0<b_1\leq \...
- 13.7k
7
votes
0
answers
179
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Differentiating $(x-1)(x-g)\dotsb(x-g^{d-1})Q(x)$
Suppose that $g$ is a nonzero, infinite-order element of a field of characteristic $0$, and $d$ is a positive integer. Let
$$ f(x):=(x-1)(x-g)\dotsb(x-g^{d-1}). $$
Suppose, furthermore, that $1<...
- 22.8k
13
votes
1
answer
522
views
Are the logarithms of the integer polynomials discrete in $L^1$ of the unit circle?
Tautologically, the integer polynomials form a discrete set in $L^1$ of the unit circle. On the other hand, a set of logarithms ordered by norm becomes generally rather denser than the original set.
...
- 13.7k
8
votes
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answers
347
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Computing coefficients of polynomials from roots in $O(n\log{n})$ time
Suppose I have a univariate polynomial $p$ over a prime-order finite field $\mathbb{F}_q$ whose roots I know.
Suppose that the roots of $p$ are always an $n$-sized subset of $R=\{1,2,\dots,N\}, N <...
- 191
2
votes
0
answers
65
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Density of integral values of a rational function
Let $\mathbf{x} = (x_1, \cdots, x_n)$, and consider a rational function $F : \mathbb{R}^n \rightarrow \mathbb{R}$ be given by
$$\displaystyle F(\mathbf{x}) = \sum_{i = 1}^m \frac{Q_i(x_1, \cdots, x_{...
- 25.5k
2
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93
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Can entropy of a network be written as a polynomial?
In my research, I met a problem here.
Consider a weighted graph Laplacian matrix
$$\mathcal{L}_w(\mathcal{G}) = DWD^T,$$ where $D$ is a incidence matrix and $W$ is diagonal with each diagonal ...
- 345
2
votes
0
answers
242
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Concerning certain Keller maps of $k[x,y]$
Let $k$ be a field of characteristic zero.
Let $(x,y) \mapsto (p,q) \in k[x,y]$ be a Keller map, namely, a $k$-algebra endomorphism of $k[x,y]$ with $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in k-\{0\}...
- 2,783
1
vote
1
answer
169
views
Explicit bivariate quadratic polynomials where Coppersmith is better than standard solver?
http://www.numbertheory.org/pdfs/general_quadratic_solution.pdf gives a general method to solve quadratic bivariate diophantine equation while Coppersmith introduced a method to solve bivariate ...
- 13.7k
5
votes
3
answers
758
views
Does there exist another form of the derivative for polynomials?
Let $F : \mathbb{R}[X] \rightarrow \mathbb R[X]$ be a linear map and let $H \in \mathbb{R}[u,x,y,z]$ be a polynomial. Suppose that
$$ F(P \cdot Q) = H(F(P),F(Q),P,Q)$$
for all $P, Q \in \mathbb{R}[X]...
- 3,801
17
votes
2
answers
1k
views
$P(x)=P(y)$ has infinitely many integer solutions
Determine all polynomials $P(x)$ with integer coefficients such that $P(x)=P(y)$ has infinitely many integer solutions in integer $x$ and $y$ with $x \neq y$.
Choose $P(x)=a_n(x-k)^{2n}+a_{n-1}(x-k)^{...
- 501
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votes
1
answer
121
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Family of zeros of polynomials
Let $k$ be an infinite field and $P(X_1,\dots,X_n)\in k[t][X_1,\dots, X_{n}]$, suppose that there exists a finite field extension $L$ of $k$ such that
$P(x'_1,\dots,x'_n)\in L[t]^{*}=L^{*}$ with $x'...
- 3,432
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0
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96
views
Multivariate polynomial with infinite but discrete roots on one variable
I want to know if there exists a polynomial $ P(z, x_1,x_2,\ldots,x_n)$ over the rationals such that the set
$$
Z_P = \{z | \exists x_1,\ldots,x_n. P(z, x_1,x_2,\ldots,x_n) = 0 \} \subsetneq \mathbb Q
...
- 163
5
votes
1
answer
143
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Elementary proof of growth estimate for a polynomial via size from its zero set
The paper Asymptotic properties of polynomials and algebraic functions of several variables by Gorin contains the following.
Lemma 3.1. Let $f\in \mathbb R[x_1,\dots,x_n]$. Suppose $f$ has a root in ...
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3
votes
1
answer
624
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Lower bound of the expectation of the product of inner products of random vectors
I encountered the following value in my research:
Let $n,m$ be some integer. Suppose $\alpha_1,\dots,\alpha_m$ are unit vectors in $\mathbb{R}^n$.
Denote
$$
L = \mathop{\mathrm{E}}_x[ \prod_{1\...
- 1,531
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votes
0
answers
75
views
Root of a polynomial and its reciprocal modulo a prime
Fix a large prime $p$ and let $k \leq 10 \sqrt{p}$. What are the $z\in \mathbb{F}_p$ that satisfy $$f(z) = f(1/z) = 0,$$ where \begin{align*}f(z) &= (2-z) \cdots (2k-z) + (-1)^{k+1} \cdot 3 \...
- 2,283
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0
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128
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Homomorphisms from $k[x,y]$ to $k[x,x^{-1},y]$
Let $k$ be a field of characteristic zero and let $R_{-1}:=k[x,x^{-1},y]$
be the $k$-algebra of polynomials in $x,y$ containing the inverse of $x$, denoted by $x^{-1}$.
Let $f: k[x,y] \to R_{-1}$ be ...
- 2,783
17
votes
1
answer
474
views
Irreducibility of root-height generating polynomial
The height $ht(\alpha)$ of a positive root $\alpha$ in a (finite, crystallographic) root system $\Phi$ is $\sum_{i=1}^n c_i$ where $\alpha = \sum_{i=1}^n c_i \alpha_i$ is its decomposition as a sum of ...
- 2,693
2
votes
0
answers
117
views
A special case of the polynomial Bézout's identity: bounding the co-factors
Let $F$ be a field of prime order $p$. Suppose that $f\in F[x]$ is a non-zero polynomial of degree $\deg f<p$. If $f$ does not have multiple roots, then there exist polynomials $P,Q\in F[x]$ such ...
- 550
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votes
1
answer
193
views
Upper bound over $[0,1] $ for strange family of polynomials
Let $n$ integer $\geq 2,$ $x$ real, and $$P_n(x)=\displaystyle \sum_{k=0}^{n-1} C_{n+k}^n (-x)^k \alpha_{n,k}$$ and where $\forall k$ such that $0 \leq k \leq n-1 $ $$ \alpha_{n,k}= \displaystyle \...
- 417
5
votes
1
answer
472
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Intuition behind the growth condition in the result of Griffin, Ono, Rolen and Zagier on Jensen polynomials
With great pleasure I read the recent paper of Griffin, Ono, Rolen and Zagier proving the surprising result that the Jensen polynomials $J^{d, n}_\alpha$ for a sequence $\alpha = \{\alpha(0), \alpha(1)...
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4
votes
0
answers
104
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Division of bivariate polynomials
The following theorem (lemma 4.2.18 on page 97) is proven in thesis "Computationally efficient Error-Correcting Codes and Holographic Proofs" by Daniel Alan Spielman:
Let $E(X, Y)$ be a polynomial ...
3
votes
1
answer
370
views
Infinite order automorphisms of planar polynomials
Let $R_n$ be the integral polynomial ring $\mathbb{Z}[x_1,x_2,...,x_n]$, let $A_n$ be the group of ring automorphisms $\mathrm{Aut}(R_n)$, and for $f\in R_n$ let $\mathrm{Aut}(f)=\{\alpha\in A_n\ |\ \...
- 8,394
16
votes
4
answers
3k
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Roots of $x^n-x^{n-1}-\cdots-x-1$
It is easy to see that $f(x)=x^n-x^{n-1}-\cdots-x-1$ has only one positive root $\alpha$ which lies in the interval $(1,2)$. But it is claimed that this root is a Pisot number (a.k.a. PV number), i.e.,...
- 527
5
votes
1
answer
287
views
Is there a general geometric characterization for polynomials to be linearly dependent?
Consider $P$ the complex projective plane, and fix a line $L$ in $P$
I had a conjecture, that prof. I. Dolgachev showed me how to prove, that $3$ quadratic polynomials depending on a variable $z \in ...
- 5,011
3
votes
1
answer
129
views
Intersection of quadratic equations with planted solutions?
Suppose we have two quadratic equations in $\mathbb R[x_1,\dots, x_n]$. What is the expected dimension of their intersection?
In general what can we say about intersection of $k$ quadratics? How many ...
- 13.7k
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votes
0
answers
515
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Intersection of an ideal and a subring
Is there a Grobner basis method that can compute the intersection of an ideal $I$ of a polynomial ring $R$ and its subring $R^\prime$? For example, I have an ideal $I=(x+y+z^2,1+xyz+yz+xz)$ of $\...
- 245
30
votes
0
answers
873
views
Three real polynomials
Theorem. Let $f,g$ be two real polynomials, and suppose that their Wronskian $W(f,g)=f'g-fg'$ has only real roots. Then on any interval $I\subset\mathbf{R}$ containing no roots of $W$ every non-...
- 88.9k
7
votes
1
answer
311
views
Taylor's polynomials and loss of real roots
Real-rootedness, log-concavity, and unimodality are intertwined properties. It's in this light that I was prompted to ask the question below.
Suppose the roots of a polynomial $p(x)$ are all real and ...
- 41.8k
1
vote
1
answer
276
views
How to find a cyclotomic polynomial of degree d that decompose into d irreducible polynomials in $Z_6$?
More specifically, I need the degree $d$ to be around 1024. I can easily find the cyclotomic polynomial of degree 1024 that satisfies the above requirement in $Z_2$, i.e., $x^{1024}+1$, which is equal ...
- 33
1
vote
1
answer
216
views
Constrained optimization of sum of squares polynomials
Consider the problem
$$
\min p(x) \text{ subject to } g_j(x)\le 0
\quad
p,g_j\in\text{SOS},
\qquad
(*)
$$
i.e. $p,g_j$ ($j=1,\ldots,m$) are sum of squares (SOS) polynomials. Can this problem be ...
- 680
3
votes
2
answers
623
views
Transformation of a fourth degree polynomial
Given $$ P (x) = ax ^ 4 + bx ^ 3 + cx ^ 2 + dx + e = a (x ^ 2 + p_1x + q_1) (x ^ 2 + p_2x + q_2) $$ for some $ a, b, c, d, e, p_1, q_1, p_2, q_2 \in \mathbb R$, prove that $ P (x) $ can be reduced to ...
- 101
4
votes
0
answers
246
views
Is every algebraic curve the critical set of an algebraic function?
Is every algebraic curve in $\mathbb{R}^2$($\mathbb{C}^2$), the set of critical points of a polynomial in $\mathbb{R}[x,y]$($\mathbb{C}[x,y]$)?
In particular what is a real (complex) polynomial whose ...
- 312
7
votes
3
answers
396
views
On $\{P(x)+Q(y):\ x,y=0,\ldots,p-1\}$ with $p$ prime
QUESTION: Is my following conjecture (formulated in 2016) true? How to solve it?
Conjecture. For any non-constant polynomials $P(x)，Q(x)\in\mathbb Z[x]$, there is a positive integer $N(P,Q)$ ...
- 14.5k
2
votes
1
answer
168
views
SDP representation of ideal polynomials for positivstellensatz refutations
If we want to certify the nonexistence of real solutions to a polynomial system of equations, i.e.
$$ S = \{ x\in \mathbb{R}^n \ | \ h_i (x) = 0, \ i=1,\dots,t \} = \emptyset, $$
we can produce a ...
- 21
3
votes
0
answers
62
views
Part 2: When does the maximal root of this polynomial have unit magnitude? Prove an inverse linear relation between parameters
Examine the polynomial
$$
x^{\tau+1}-\left(1+m\right)x^{\tau}+mx^{\tau-1}+\left(1-m\right)\alpha=0\,.
$$
with positive parameters $\tau,\alpha$ and $m<1$, and denote $\left|x_{\max}\left(\tau,\...
- 876
4
votes
1
answer
125
views
When does the maximal root of this polynomial have unit magnitude? Prove an inverse linear relation between parameters
Examine the polynomial
$$
x^{\tau+1}-x^{\tau}+\alpha=0\,
$$
and denote by $\left|x_{\max}\left(\tau,\alpha\right)\right|$ as the
maximal magnitude of a root of this equation. For $\tau>1$, I ...
- 876
2
votes
1
answer
141
views
Reading off top hook-lengths in partitions
Given an integer partition $\lambda$ and its Young diagram $Y_{\lambda}$, let $h_{\lambda}(i,j)$ stand for the corresponding hook length of the cell $(i,j)\in Y_{\lambda}$. Write $\lambda\vdash n$ for ...
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