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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Computer algebra programs that can solve polynomial systems on algebraically closed fields besides MAGMA

I was wondering which computer algebra programs out there can solve polynomial systems on the algebraic closure of $\mathbb{Q}$ analytically and efficiently. So far, I only found MAGMA with its ...
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3 votes
1 answer
79 views

Expressing a vector valued function in terms of its derivatives

Consider a function $$ f:\mathbb{R}^n\rightarrow\mathbb{R}^m $$ given by $m$ functions $f_i:\mathbb{R}^n\rightarrow \mathbb{R}$ that we can assume to be polynomials in $x_1,\dots,x_n$. Does there ...
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13 votes
2 answers
1k 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 ...
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1 vote
0 answers
68 views
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Conjectures inspired in the context of Casas-Alvero conjecture, via the logarithmic derivative of derivatives of a polynomial

In the post (cross-posted in Mathematics Stack Exchange with identificator MSE 4244256 and same title) we assume that $P(x)=a_0+a_{1}x+\ldots+a_{n-1}x^{n-1}+a_{n}x^n$ is a polynomial of degree $1<\...
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4 votes
0 answers
126 views

Explicit expression for recursive sums - II

A twist on just unfolded recursive summation formula. Let polynomials in nonnegative integer variables $t_1,t_2,\dots$ be defined by the recurrence: \begin{split} g_0 &= 1, \\ g_k(t_1,t_2,\dots,...
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5 votes
1 answer
151 views

Is it possible to solve sextic equations using the Fox H function?

Although the Kampé de Fériet function can solve the sextic equation, the details about it are shrouded in the fog of more than a century ago. In contrast, we know more about the Fox H function, and we ...
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9 votes
2 answers
384 views

Explicit expression for recursive sums

Let $t_1,t_2,\dots,t_k$ be non-negative integers. Can the following sum $$f_k(t_1,t_2,\dots,t_k):=\sum_{j_1=0}^{t_1} \sum_{j_2=0}^{t_2+j_1} \sum_{j_3=0}^{t_2+j_2} \dots \sum_{j_k=0}^{t_k+j_{k-1}} 1$$ ...
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1 vote
1 answer
290 views

Square root of prime numbers

I found that the square root of any prime number S can be approximated, at the n-th order, as a rational number represented by the polynomials shown below. $x_0$ is an initial seed, which is a ...
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3 votes
0 answers
35 views

Derivative of characteristic polynomial of a graph and derivative of characteristic polynomial of a vertex-deleted subgraph have a common root

Let $G$ be a simple graph and $G-i$ be one of its vertex-deleted subgraphs. Let $\phi(G,x)$ and $\phi(G-i,x)$ be the characteristic polynomials of $G$ and of $G-i$ respectively, with respect to their ...
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Generalization of "Lagrange interpolation" over non-division rings

The theorem below is from pages 4 and 5 in Singmaster - On polynomial functions $\pmod m$ (Theorem 10) on polynomials in $\mathbb{Z}_m[x]$. Let $f$ be a polynomial function $\pmod{m}$. Then $f$ has a ...
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1 vote
0 answers
88 views

Clarification about theorem on vanishing polynomials

The theorem below is from page 3 in the this paper on polynomials in $\mathbb{Z}_m[x]$. Let $F$ be a polynomial in $\mathbb{Z}_m[X]$. Then $f \equiv 0$ iff $$F \equiv F_nS_n + \sum_{k=0}^{n-1}a_k(m/(...
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2 votes
0 answers
57 views

What's the number of irreducible polynomials over rational numbers with constraint on coefficients?

Suppose you have polynomials $f(x) = \sum\limits_{k=0}^n a_k x^k$ where $f \in \mathbb{Q}[x]$ and $a_k$ can be either $0$ or $1$. There are $2^n$ such polynomials in total. How can I find a number of ...
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3 votes
0 answers
118 views

Order of elements $\gamma$ and $1-\gamma$ in $\mathbb{F}_q$

I'd like to find if possible the orders possible for $1-\gamma$ given a $\gamma\in\mathbb{F}_q$ of given order $\mathcal{o}$ (where $q=p^f$). Which is clear is that these order have the same degree (...
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5 votes
2 answers
298 views

Writing $1-xyzw$ as a sum of squares

Can you write $1 - xyzw$ in the form $p + q (1 - x^{2}-y^{2}-z^{2}-w^{2})$ where $p$ and $q$ are polynomials that are of the form $\sum g_{i}^{2}$ where $g_{i}$ $\in$ $\mathbb{R}[x,y,z,w]$? For ...
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4 votes
1 answer
318 views

On the solvability of the equation $ax^p+bx^{p-1}+cx+d=0$ by radicals

Joint with Qing-Hu Hou at Tianjin Univ., we seek for explicit criteria via coefficients for the solvability of an algebraic equation by radicals. In this direction, we formulate the following ...
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Algebraic relations between the symmetries $f(x,y),f(y,z), f(z,x)$

When $f(x,y)=x-y$ the map $$(a,b,c)\mapsto (f(a,b),f(b,c),f(a,c))$$ gives a parametrization of $x+y=z$. More generally, for an arbitrary polynomial $f$ in $k$ variables and an integer $n$ bigger than $...
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3 votes
1 answer
263 views

A criterion for the equation $ax^n+bx+c=0$ not solvable by radicals via $a,b,c$ and $n$

Galois revealed that an algebraic equation $f(x)=0$ with coefficients in a field $K$ of zero characteristic is solvable by radicals if and only if the Galois group of $f(x)$ over $K$ is solvable. ...
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5 votes
1 answer
226 views

Is the minimal polynomial of an algebraic formal Laurent series always separable?

Let $f(x)\in K((x))$ be an algebraic formal Laurent series and let $P(x,y)\in K(x)[y]$ be its minimal polynomial. Is $P(x,y)$ always separable? An example of non separable polynomial comes from ...
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5 votes
1 answer
299 views

Counting monomials and the Catalan numbers

Given a multi-variable polynomial $F$, denote the number of monomials by $N(F)$. Take for instance, \begin{align*}N(x(x+y)+(x+y)^2-(x-y)^2)=N(x^2+5xy)&=2 \qquad \text{and} \\ N((x+z)(x+y)^2)=N(x^3 ...
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1 vote
1 answer
76 views

Order of roots for a polynomial $P\in\mathbb{F}_p[T]$

Let $P\in\mathbb{F}_p[T]$ (not supposed irreducible). All roots $\xi$ of $P$ have a certain order $k$ such that $\xi^k=1$. Question: is it possible to know the order of the roots of the given ...
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5 votes
2 answers
394 views

When do multiple polynomials have a common root?

I was wondering if it is well understood under what circumstances say three univariate polynomials $f(x),g(x),h(x)$ have a common root. In this situation, I can see that the resultant of each pair ...
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1 vote
1 answer
134 views

roots of a fourth degree polynomial function (Vieta) [closed]

Question I am interested in the root of the polynomial function : $x^4+(a+b+c+d-2)x^3+(ab+ac+ad+bc+bd+cd-2b-2c-a-d)x^2 +(abc+abd+acd+bcd-ab-ac-ad-2bc-bd-cd-a-d+b+c)x+ abcd-abc-bcd-ad+bc=0$ Under the ...
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2 votes
1 answer
154 views

On roots of irreducible quadratics modulo composites

Assume factorization of $N$ is unknown. What is the best complexity we know to find roots of the irreducible equation $$ax^2+bx+c\equiv0\bmod N?$$ Is this problem equivalent to any hardness results?
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1 vote
1 answer
69 views

Complex polynomial-like functions with conjugate terms

Is there study on polynomial-like functions of the following kind? $$f(z) = c_0 + a_1z+b_1\bar{z} + a_2z^2+b_2\bar{z}^2 + ...+ a_nz^n+b_n\bar{z}^n$$ My reason for studying it is polynomials are ...
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2 votes
0 answers
98 views

When $\gcd(P(x),Q(x))\bmod R(x)=\gcd(P(x) \bmod R(x),Q(x) \bmod R(x))$?

Let's $P,Q\in\mathbb K[x]$, with $\mathbb K$ a finite field. On what necessary and sufficient condition on $R \in \mathbb K[x]$ is it : $\gcd(P(x),Q(x))\bmod R(x)=\gcd(P(x) \bmod R(x),Q(x) \bmod R(x))...
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3 votes
1 answer
364 views

Root of polynomials in a finite field

I am looking for a way to find out if a polynomial $P\in \mathbb Z/p\mathbb Z=\mathbb F_p$, of great degree, has roots in $\mathbb F_p$, with $p$ a big prime number. For example : $p=2^{2020}-69$ ...
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  • 3,013
2 votes
3 answers
375 views

Useful software for variable elimination

I have three non-homogeneous trivariate polynomials in $\mathbb Z[x,y,z]$ and I want to eliminate the variables $y$ and $z$ to get a polynomial in $x$. The monomials of the polynomials are $\{1,x^4,x^...
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9 votes
1 answer
823 views

Impact of Ramanujan's Note on a set of simultaneous equations

I had been pointed to Ramanujan's 1912 article Note on a set of simultaneous equations in this answer to my former question about the Solvability of a system of polynomial equations. While the ...
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2 votes
0 answers
368 views

Minimal number of generators of kernel of a matrix of polynomials over $\mathbb Q$ and $\mathbb Z$

$\DeclareMathOperator\im{im}$I have asked some related questions before on math.SE here and on MathOverflow here (answered here). This post is self-contained. Let $R' = \mathbb Q[x_1,\dotsc,x_n]$, and ...
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1 vote
0 answers
18 views

Minimum eigenvalue of normal matrix with polynomial basis

For each $n\in \mathbb{N}\cup\{0\}$, let $x_n(t)=\frac{t^n}{n!}$ for all $t\in [0,1]$. As the functions $X_N=(x_0 ,\ldots, x_N)$ are linearly independent, the matrix $ \int_0^1 X_N(t)^\top X_N(t)\,\...
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7 votes
1 answer
365 views

Factorization of an irreducible polynomial in the field extension it defines

In field theory, the following fact is used in the construction of splitting fields: Given a field $F$ and an irreducible polynomial $f \in F[x]$, the quotient $F[\alpha]/(f(\alpha))$ is a field ...
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1 vote
1 answer
233 views

Constructing a free resolution of a $\mathbb Z[x_1,\dotsc,x_n]$-module using a related free resolution of a $\mathbb Q[x_1,\dotsc,x_n]$-module

I have asked a related question on math.SE here, but the notation is a bit different. As the title says, I am interested in constructing a finite free resolution of a $\mathbb Z[x_1,\dotsc,x_n]$-...
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2 votes
2 answers
447 views

About roots of polynomials [closed]

Let $n\in\mathbb N^*$, $P(x)=a_0+\dotsb+a_{n-1}x^{n-1}+x^n$ and $r_1,\dotsc,r_n\in\mathbb C$ the roots of $P$. Is it true $\lim\limits_{\max(\lvert a_i\rvert,i=0\dotsc n-1)\rightarrow 0} \max(\lvert ...
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11 votes
2 answers
831 views

Polynomials for which roots can be expressed as polynomials in a single root

Classical Galois theory gives necessary and sufficient conditions for the roots of a polynomial in $k[x]$ to be expressible in terms of nested radicals of the coefficients. Suppose instead that a ...
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2 votes
1 answer
119 views

Is a finite order automorphism of k[x,y] necessarily linear?

Let $k[x,y]$ be the polynomial ring in two variables over a field $k$ of characteristic zero. Every $k$-algebra automorphism of $k[x,y]$ is tame (e.g. the paper of McKay and Wang). It was pointed out ...
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3 votes
1 answer
214 views

Analytic expression for the coefficient of a multivariate polynomial

Does there exist some method for finding an analytic expression for the coefficient of $z_1^kz_2^kz_3^k$ in: $$[(1+z_1)(1+z_2)(1+z_3)(1+z_1z_2)(1+z_1z_3)(1+z_2z_3)(1+z_1z_2z_3)]^{k}$$ or is it ...
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2 votes
1 answer
188 views

Singular locus of the discriminant variety

I asked this question on MSE some time ago but didn't get a response. Everything can be assumed in $ \mathbb{C} $, or atleast in characteristic $ 0 $. Consider degree $ d $ hypersurfaces in projective ...
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1 vote
0 answers
42 views

Strong Rayleigh measures conditioned on partial sum

Consider a binary random vector $X=(X_1,\ldots,X_n)$ with a strong Rayleigh distribution (i.e., its multi-affine generating polynomial is stable). It is well known that the law of $X$ remains strong ...
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0 votes
1 answer
136 views

Restrictions on exponents in multinomial formula

From the multinomial formula we have $$(x_1 + x_2 + \dotsb + x_m)^n = \sum_{k_1+k_2+\dotsb+k_m=n, \ k_1, k_2, \dotsc, k_m \geq 0} {n \choose k_1, k_2, \dotsc, k_m} \prod_{t=1}^m x_t^{k_t}\,.$$ I ...
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22 votes
2 answers
2k views

Can one deduce the fundamental theorem of algebra from real calculus and linear algebra?

Motivation: let $A\in\mathbf{R}^{n\times n}$ be symmetric. Then by the method of Lagrange multipliers, a maximum of $x\mapsto x^tAx$ on the compact unit sphere $\mathbf{S}^{n-1}$ must be an ...
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0 votes
0 answers
69 views

Low degree factors of $f_1(x)^n+f_2(x)^n-f_3(x)^n$

Let $f_1(x),f_2(x),f_3(x)$ be coprime polynomials with integer coefficients. Let $g(x)$ be a factor of $f_1(x)^n+f_2(x)^n-f_3(x)^n$. We are interested when the degree of $g(x)$ can be low. One ...
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7 votes
2 answers
336 views

Prove or disprove that the power of positive term polynomial will be eventually single peak

This is a question that a classmate asked me three years ago. Let $P(x)=\sum_{i=0}^n a_ix^i$ be a polynomial such that each $a_i>0$. Prove or disprove that there exists a positive integer $r$ such ...
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  • 509
4 votes
1 answer
221 views

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 ...
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0 votes
0 answers
31 views

Proving sup-norm of a specific polynomial (Kaisjer--Varopoulos, 1974)

This question regards a proof in the addendum (due to Kaisjer and Varopoulos) to "On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory,&...
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4 votes
0 answers
104 views

$P \circ P \in \mathbb{Z}[X]$ and $P \circ P \circ P \in \mathbb{Z}[X]$ [duplicate]

Let $P \in \mathbb{Q}[X]$, is it true that $P \circ P \in \mathbb{Z}[X]$ and $P \circ P \circ P \in \mathbb{Z}[X]$ implies $P \in \mathbb{Z}[X]$ ? The two conditions are necessary because, if $P=2X^2-\...
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4 votes
1 answer
157 views

Positivity of real functions in two variables

Assume that $f_0,f_1,f_2$ are polynomial functions of degree two in two variables. This means that the $f_i$ are linear combinations with real coefficients of $x^2,xy,x,y^2,y,1$. Consider the function ...
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4 votes
1 answer
261 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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0 votes
0 answers
58 views

Irreducibility of cyclotomic polynomial with change of variable

Consider the cyclotomic polynomial $\Phi_k(x - \alpha)$ where $\alpha$ is an algebraic integer in a number field $\mathcal{K}$. Is $\Phi_k(x - \alpha)$ irreducible in $\mathcal{K}[x]$ ? If yes, also $\...
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  • 19
7 votes
2 answers
773 views

Product of complex numbers on the unit circle with largest real part

Let $T = \{z_1, \ldots z_n\}$ be a finite set of complex numbers on the unit circle. I would like an algorithm which can quickly compute the nonempty subset $S \subset T$ which maximizes $$\left| \...
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11 votes
2 answers
484 views

Undecidability of irreducibility of infinite families of integer polynomials?

A recent question, Is irreducibility of polynomials $\in\mathbb{Z}[X]$ over $\mathbb{Q}$ an undecidable problem? was quickly answered in the negative. I am wondering if there is a simple example of a ...
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