Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

1
vote
0answers
42 views

Formula for the index of regularity of a generic Hilbert function

Is there an explicit formula for the index of regularity of a generic Hilbert function in two variables? (i.e., the Hilbert function of an ideal of $k[X,Y]$ generated by $r$ generic forms $f_{i}$ of ...
6
votes
0answers
154 views

Infinitude of cyclotomic polynomials with a certain number of terms

Let $\Phi_n$ be the $n$th cyclotomic polynomial: $${\Phi _{n}(x)=\!\!\prod _{\substack {1\leq k\leq n \\ \gcd(k,n)=1}} \!\!\big(x-e^{2i\pi {k/n}}\big).}$$ Here is a list of the first 30 cyclotomic ...
3
votes
1answer
121 views

Trees and Shabat polynomials

Recently, I read the relation between Shabat polynomials and trees. The book [0] says that if $p: \mathbb{C} \to \mathbb{C}$ is a degree-$n$ polynomial such that the segment $[-1,1]$ contains no ...
6
votes
2answers
487 views

Kernel of evaluation map into field of quotients

Let $R$ be an integral domain and for $a \in R$ denote by $\text{eval}_a: R[X] \to R$ evaluation at $a$. It's well-known (and easy to see) that $$\ker(\text{eval}_a)=(X-a).$$ The next more ...
3
votes
0answers
127 views

Reverse Markov-Bernstein inequality for trigonometric polynomials

Let $r(t)$ be a real trigonometric polynomial of degree $n>1$. Assume it has zero at $t=0$ of multiplicity $k>0$. What can be said about the lower bound of the constant $c(k,n)$ such that $$ \...
6
votes
3answers
394 views

Sufficient conditions for a polynomial to be reducible over the integers

There are several well-known criteria for a polynomial with integer coefficients to be irreducible over $\mathbb{Z}$, e.g., Eisenstein's criterion. I'm looking for the opposite: other than ...
1
vote
1answer
48 views

For univariate polynomial is non-negativity on an interval equivalent to having a nonnegative scalar product with non-negative polynomials

Let $\mathbb R_d[t]$ be the set of univariate polynomials in the variable $t$ of degree $d$, and $S$ be the set of elements of $\mathbb R_d[t]$ that are nonnegative on $[0, 1]$. Does the following ...
33
votes
7answers
1k views

On the polynomial $\sum_{k=0}^n\binom{n}{k}(-1)^kX^{k(n-k)}$

Let $n = 2m$ be an even integer and let $F_n(X)$ be the polynomial $$F_n(X):=\sum_{k=0}^n\binom{n}{k}(-1)^kX^{k(n-k)}.$$ I observed (but cannot prove) that the polynomial $F_n$ is always divisible by $...
7
votes
1answer
172 views

Discriminant of numerator of inverse logarithmic derivative operator iteration

Let $T:\mathbb Q(x)\to \mathbb Q(x)$ be the operator of inverse logarithmic derivative, i.e. $$Tf=\frac{f}{f'}.$$ Define $$p_n(x)=T^n\left(x-\frac{x^2}{2}\right).$$ Let $f_n(x) \in \mathbb Z[x]$ be ...
1
vote
0answers
75 views

Radial similarity of Newton polytopes

Let $k$ be a field of characteristic zero, and assume that $p,q \in k[x,y]$ is a Jacobian pair, namely, $p_xq_y-p_yq_x \in k^*$ (= the determinant of the Jacobi matrix $\in k^*$). It is known that ...
1
vote
1answer
143 views

Is there always a polynomial with real zeroes between two polynomials with real zeroes?

Suppose that we have two complex polynomials $p(z)=\sum_{k=0}^n p_kz^k$ and $q(z)=\sum_{k=0}^n q_kz^k$ and also that we have $|p_k|<|q_k|$ for $k=0,1,...,n$. We say that a polynomial $r$ is ...
0
votes
0answers
81 views

A simple question about polynomial ideals

Let $J$ be a homogeneous ideal of $R=\mathbf{C}[X_{1},...,X_{N}]$ and let $d$ and $m$ be positive integers with $d>m$. Suppose (H) $J_{d}+\langle f\rangle_{d} \neq R_{d}$ for every $f\in R_{m}$, ...
0
votes
1answer
114 views

Polynomials $p$ such that $p$ and $p'$ preserve nonnegative numbers

Expanding on a previous post I made recently, let $$ \mathscr{P}:= \{ p(x) \in \mathbb{R}[x] \mid p(x) \ge 0,~\forall x\ge 0\}. $$ The Pòlya-Szegö theorem (see Theorem 3.21 here) asserts that $p \in \...
0
votes
1answer
833 views

What is special about 2 + $\sqrt{3}$?

Well, one thing is special about it, but it takes a while to explain. Please let me know, whether this number occurs in other special occasions as well. The explanation: Let $p$ be a complex ...
3
votes
0answers
77 views

Deterministic procedure to find irreducible polynomials

In $\Bbb F_q[x_1,\dots,x_n]$ given $d_1,\dots,d_n\in\Bbb N$ is there a deterministic $O(poly(nd\log q))$ algorithm to find an irreducible polynomial with $d=\max_{i\in\{1,\dots,n\}}d_i$ and $d_i=deg(...
5
votes
0answers
151 views

Cardinality of the image of a polynomial modulo $p^n$

Let $f \in \mathbb{Z}[x]$ be a nonconstant polynomial and let $p$ be a prime number. I'm looking for results about $$N_f(p^k) := \#\{(f(n) \bmod p^k) : n \in \mathbb{Z}\},$$ as $k \to +\infty$, where $...
0
votes
0answers
64 views

A system of polynomial equations has exactly one positive real zero

Recently I consider the following system of polynomial equations: \begin{equation} \sum_{i=1}^m c_i(\boldsymbol{\alpha}_i-\boldsymbol{\beta}_r)\mathbf{x}^{\boldsymbol{\alpha}_i}-\sum_{j=1}^{r-1}d_j(\...
2
votes
2answers
106 views

Given a polynomial system, determine a simple set containing all its solutions

Suppose we have a system of complex polynomials $f = (f_1, \ldots, f_n)$, where each $f_i$ can be viewed as a function $\mathbb{C}^n \to \mathbb{C}$. The solutions of $f$ are the points $x \in \mathbb{...
4
votes
1answer
299 views

Integral domain over which any non-constant, one variable, irreducible polynomial has degree 1

Let $R$ be an integral domain such that every non-constant, irreducible polynomial $f(X) \in R[X]$ has degree $1$. Q. is it true that $R$ is a field? If $0 \ne a \in R$ , then $X^2-a$ is ...
11
votes
2answers
555 views

Polynomials that preserve nonnegativity

A polynomial $p \in \mathbb{R}[x_1,\dots,x_n]$ is said to be positive on a subset $S$ of $\mathbb{R}^n$ if $p(x) > 0$ for every $x \in S$. The polynomial $p$ is called nonnegative if $p(x) \ge 0$ ...
12
votes
1answer
262 views

Factorization of polynomials into “shortest possible” factors

A while ago I asked a question at Mathematica.SE about how to factorize a polynomial into terms with as few monomials as possible each. I now realized that I actually do not know what is rigorous ...
4
votes
1answer
153 views

Combinatorial formula for Betti numbers of a $k[x,y]$-module

Suppose I have a (finitely-presented, say) graded module $M$ over $k[x,y]$, and I happen to know the rank $R_{(a,b),(c,d)}$ of each map $x^{c−a}y^{d−b}:M_{a,b}→M_{c,d}$ for each pair of integers with $...
26
votes
3answers
2k views

All polynomials are the sum of three others, each of which has only real roots

It was asked at the Bulletin of the American Mathematical Society Volume 64, Number 2, 1958, as a Research Problem, if a Hurwitz polynomial with real coefficients (i.e. all of its zeros have negative ...
5
votes
2answers
255 views

An equality relation for complex numbers off the nonnegative real axis [closed]

For every complex number $z$ off the nonnegative real axis there exist positive numbers $p_0,... ,p_n$ such that $\sum_{i=0}^n p_iz^i = 0$. Finding difficulty in proceeding with the problem. Need ...
20
votes
1answer
569 views

Are $\mathbb C$ , $\mathbb C[X]$ definable in $\mathbb C[[X]]$?

Let $L$ be a first-order language and $M$ be an $L$-structure. Let $D \subseteq M^n$ . Let us say $D$ is definable in $M$ if for some finite set (possibly empty) $A=\{a_1,...,a_m\} \subseteq M$ and ...
2
votes
1answer
518 views

Does anyone recognize these polynomials? Need to compute a riemann lebesgue type limit

These polynomials show up naturally in my work $$ p_n(x) = \sum_{j=0}^n {n \choose j} \frac{(-x)^j}{j!} $$ Does anyone know recognize if they belong to any class of well known polynomials. I am trying ...
4
votes
2answers
170 views

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\...
5
votes
0answers
135 views

Resultant of a binomial and a trinomial

Does anyone know of any papers dealing with the resultant of a binomial $x^n+a$ and a trinomial $x^r+bx^s+c$ ? Even special cases would be of interest. (The resultant of two binomials is well known.)...
-2
votes
1answer
68 views

What is an algorithm for generating a set of null (spacetime) vectors that add to zero? [closed]

I am interested in generating a list of n 4-vectors (t,x,y,z) such that -t^2+x^2+y^2+z^2=0 for each vector and the sum of the n 4-vectors equals zero. All of the t,x,y,z are real. I, particular, I am ...
3
votes
1answer
172 views

Closure of polynomials of a function in $L^2$

Suppose $f \colon I \to \mathbb{R}$ is a function in, say, $L^\infty$, and $I \subset \mathbb{R}$ is a bounded interval. We may assume further regularity on $f$, such as Lipschitz continuity or strict ...
1
vote
1answer
129 views

Efficient algorithm for $x^n-x \mod P(x)$ over $GF(2^{12})$

My goal is to generate an irreducible polynomial over $GF(2^{12})$ with degree $t$, which can get fairly big, let's say up to $t=200$ or so. I've found this very helpful paper that walks me through ...
2
votes
0answers
111 views

Lifting irreducibility of polynomials over an integral domain to the quotient field

Suppose $R$ is an integral domain with quotient field $K$ and $f \in R[x]$ is an irreducible polynomial. Under what conditions on $R$ and $f$ can we conclude that $f$ is irreducible in $K[x]$? It is ...
14
votes
1answer
387 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 $$f(x) = \frac{(x-a)(x-b)}{n(n-1)} f''(x).$$ ...
7
votes
0answers
147 views

Partition the rationals with respect to a multivariate polynomial which sends classes to classes

Let $R$ be a commutative ring and let $f\in R[x_1,x_2,\cdots,x_{n-1}],n\geq 2$ be a polynomial. Definition: We say $f$ is $n$-severable over $R$ if there exists a partition (of set) $$R=\coprod_{i=...
1
vote
0answers
106 views

Question regarding the image of a polynomial map containing a small box

I have the following question, which intuitively seems it should be true but I wasn't sure how to prove it rigorously. Let $\delta, \varepsilon > 0$. Let $\Psi_i(w_1, w_2, \mathbf{v})$ be a ...
0
votes
0answers
234 views

On polynomials that divide $x^m-1$?

Are there infinitely many integers $n>0$ at which a $0$ and $\pm1$ coefficient polynomial $f(x)$ that divides $x^{m}-1$ (thus $f(x)$ is product of cyclotomials) with degree $m-O(n^{})$ where $m=\...
-2
votes
1answer
124 views

Is there any pseudoprime that pass this test above tested range, or any prime that does not show these ending patterns?

if the recurrence sequence is defined by the following foormula, $d_{n + 3} = 3d_{n + 2} - d_{n + 1} - 2d_n$ where $d_1 = 1, d_2 = 3$ and $ d_3 = 7$, this produce the following complex sequence $$1, ...
4
votes
1answer
122 views

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 ...
1
vote
1answer
109 views

Locus of roots of all convex combinations of two monic polynomials, II

This post contains a revised conjecture to a conjecture I posed previously which was shown to be false. Let $p, q \in \mathbb{C}[t]$ be two monic polynomials of degree $n \ge 1$. For $\alpha \in [0,1]...
5
votes
2answers
373 views

is there any non prime (pseudoprime) that holds true for this test, or any prime that fall out of this test?

The sequence is defined by the following formula $d_{n + 3} = d_{n + 2} + 2d_{n + 1} - d_n$ where $d_1 = 0, d_2 = 1, d_3 = 2$, $\{0, 1, 2, 4, 7, 13, 23,42, ...\}$ if this sequence is calculated over ...
0
votes
0answers
32 views

Root separation of polynomials with the same coefficients

Consider two polynomials of the following form: $$f(x)=k_1 x^{a_1}+\ldots+k_d x^{a_d}$$ $$g(x)=k_1 x^{b_1}+\ldots+k_d x^{b_d}$$ with the following properties: $k_1>0$ For every $i$, if $k_i>0$ ...
4
votes
1answer
163 views

Second order recurrence relation for third order polynomial root

Consider this recurrence relation: $$ \begin{eqnarray*} f_0&=&1\\ f_n&=& \sum_{m=0}^{n-1} \frac{\left(\frac{m+3}{2}\right)_{m-1}}{\left(\frac{m+2}{2}\right)_m} f_{n-m-1} f_m\ \ \ \...
18
votes
2answers
665 views

Zeros of MacLaurin polynomials for the exponential function

Asked but never answered at MSE. Let $\exp_n(z)$ denote the nth degree Taylor polynomial of $e^z$ : $\exp_n(z) = 1 + z + z^2/2! + ... + z^n/n! \;$ . The zeros of $\exp_n(z)$ were studied by ...
1
vote
0answers
162 views

What is known about the coefficients of high powers of the Vandermonde determinant?

This question is about high powers (parameter $p$ going to infinity) of the Vandermonde determinant with $N$ entries, for fixed $N$. I expand $$ \Delta(X_1, \dots, X_N)^p = \prod_{i < j} (X_i - X_j)...
1
vote
0answers
76 views

Does there exist a triplet of polynomials with natural coefficients with the following properties?

Does there exist a triplet $(f,g,h)$ of polynomials with non-negative integer coefficients $a_i,b_i,c_i$, for chosen degree $p\gt1$ $$f(x)=a_{p}x^{p}+a_{p-1}x^{p-1}+\dots+a_{1}x^{1}+a_{0}$$ $$g(x)=b_{...
3
votes
1answer
157 views

How does the minimal degree of a monic polynomial with all values divisible by $p^n$ asymptotically behave?

Let $p$ be a prime number. For every $n \in \mathbb N$, let $A_{p,n}:=\{\deg P(X) : P(X)\in \mathbb Z[X]$ is monic and $p^n|P(m), \forall m \in \mathbb Z$ $\}$ . As user abx notes below, $A_{p,n}$ ...
13
votes
1answer
426 views

Minimal cardinality of a field where a polynomial has a root

Let $P(n)$ be the set of all monic polynomials of degree $n$ with integer coefficients, such that all coefficients have absolute value at most $2^n$. Given a positive integer $n$ let us define $A(n)$ ...
16
votes
1answer
468 views

Cyclotomic polynomials 2

My naive question may actually lead to something interesting. Let $\Phi_m(x)$, $\Phi_n(x)$ be cyclotomic polynomials, $m<n$. These polynomials are relatively prime and so there are polynomials $...
5
votes
1answer
562 views

Cyclotomic polynomials.

Let $\Phi_m(x)$ and $\Phi_n(x)$ be two different cyclotomic polynomials. Then $\Phi_m(x)$ and $\Phi_n(x)$ are coprime, so there are two polynomials $s(x), t(x)$ with, say, rational coefficients such ...
1
vote
1answer
175 views

On largest degree of polynomial related to cyclotomic polynomials - I

We know cyclotomic polynomials $\Phi_{2^kp^rq^m}(x)$ have coefficients in $\{0,\pm1\}$. What is the largest degree $f_{d,n}(x)=\frac{x^{2^{k}p^{r}q^{m}}-1}{\Phi_d(x)}$ with $\{0,\pm1\}$ coefficients ...