**0**

votes

**0**answers

71 views

### On degrees of polynomials with matching zeros in a subset

Let $S\subsetneq \Bbb R^n$ such that $|S|<\infty$ and for all partitions $S_1$ and $S_2$ of $S$ there exits a multilinear polynomial $h$ such that $h(s)=1-h(s'),\mbox{ }\forall s\in S_1\mbox{ and ...

**2**

votes

**0**answers

31 views

### Estimating polynomial approximation error in high dimension

Question
Let $x \in [-1, 1]^d \subset \mathbb{R}^d$ be a $d$-dimensional variable and assume that -- given $n$ -- I have a way of computing a polynomial $p_n(x)$ of degree $n$ that approximates a ...

**-2**

votes

**0**answers

118 views

### how to solve 3 6-degree polynomial equations for 3 variables? [on hold]

I am a physicist and need to solve three $6$-order polynomial equations for $3$ unknowns $(p, q, r)$. Here is the system of equations looks like:
$$\sum(A[n]*p^i*q^j*r^k) = 0,$$
...

**-2**

votes

**0**answers

78 views

### X^2e + X^e + 1 is irreducible for e which is a power of 3? [on hold]

I am looking for a proof or reference to the fact that the trinomials of the form
X^2e + x^e + 1
Are irreducible in GF(2) for e which is a power of 3.
Please help!
Lear

**2**

votes

**0**answers

114 views

### bound on degree of certain polynomials

Consider $m$ polynomials $f_i$ for $i=\{1,\dots,m\}$ in $\Bbb R[x_1,\dots,x_n]$ of degree $d_i$ such that $$Z(f_i)\bigcap Z((p))\neq \emptyset$$ for the sphere $p$ passing through $\{0,1\}^n$. What is ...

**5**

votes

**1**answer

345 views

### degree of polynomials in nullstellensatz

$(A)$ If $K$ is algebraically closed field, then consider $m$ polynomials $f_i$ for $i=\{1,\dots,m\}$ in $K[x_1,\dots,x_n]$ of degree $d_i$ each with no common root. We know there exists $g_i$ for ...

**1**

vote

**0**answers

58 views

### Identity of Bernoulli polynomials

consider the Bernoulli polynomials defined by the generating function:
$$\left(\prod_{i=1}^m \frac{a_i}{\left( e^{a_i}-1 \right)}\right)e^{xt}=\sum\limits_{n=0}^{\infty}B^{m}_n\left(x\vert ...

**3**

votes

**1**answer

237 views

### Sum of two squares and implication of Bunyakovsky conjecture

Bunyakovsky conjecture states that a polynomial with integer
coefficients takes infinitely many prime values at integers,
unless this is impossible for trivial reasons.
Let $a_1(x), a_2(x), a_3(x), ...

**1**

vote

**1**answer

129 views

### Reducible polynomials

Say one only seeks to identify whether a given polynomial over $\mathbb{Z}[x]$ is reducible, then what are the best ways known to solve this?
$(1)$ If reducible, the algorithm should correctly say ...

**1**

vote

**0**answers

63 views

### On reducible polynomials

Let $f(x),g(x)\in\Bbb Z[x]$ with $deg(f)>deg(g)$.
Given an integer $B$, is there any algorithm that runs in $\log^c |B|$ for some fixed $c\in \Bbb R$ to find a $h(x)\in\Bbb Z[x]$ (if one exists) ...

**1**

vote

**0**answers

68 views

### Does this condition imply a polynomial is a product of linear factors

Let $\Lambda$ be a lattice (i.e. $\Lambda \simeq \mathbb{Z}^n$) with a positive subcone $\Lambda^+$. Let $H: \Lambda^+ \rightarrow \mathbb{C}$ be a function such that $\forall\mu \in \Lambda^+$, ...

**5**

votes

**1**answer

421 views

### Disjoint images of polynomials

Are there any $f,g \in \mathbb{Q}[x]$ such that for every root of unity $\zeta$, and every $a,b \in \mathbb{Q}(\zeta)$, $f(a) \neq g(b)?$

**1**

vote

**0**answers

42 views

### Name for generalization of bivariate weighted-homogeneous polynomials

A polynomial $f = \sum_j c_j X^{\alpha_j}Y^{\beta_j}\in\mathbb K[X,Y]$ is said weighted-homogeneous if there exist $p$, $q$ and $d$ (where $p$ and $q$ are not both $0$) such that ...

**4**

votes

**1**answer

76 views

### Orthogonal polynomial under linear transformation

Let $M_n(x) = x^n$ be the standard monomials. The binomial formula allows one to expand $M_n(ax+b)$ as a linear combination of $M_k(x)$, for $k \leq n$, giving
$$
M_n(ax+b) = (ax+b)^n = \sum_{k=0}^n ...

**0**

votes

**1**answer

78 views

### Integer-valuedness of a polynomial determined by output of first n integers? [closed]

An integer-valued polynomial is a polynomial $p(x)$ such that $\forall x \in \mathbb{Z}, p(x) \in \mathbb{Z}$.
Theorem: For any $n$-degree polynomial $p$, if $p(x) \in \mathbb{Z}$ for all $x \in \{0, ...

**1**

vote

**0**answers

33 views

### Distributing partially known data between n parties

Assume that $n = 2r+1$. There are $n$ elements $a_1,a_2,\ldots,a_n$ from a finite field $\mathcal{F}$, and $n$ parties. Each party knows the values of at least $r+1$ elements out of those $n$ ...

**14**

votes

**2**answers

872 views

### Images of polynomials

Let $f,g \in \mathbb{Q}[x]$ be polynomials such that $\{f(a) : a \in \mathbb{Q}\} \subseteq \{g(a) : a \in \mathbb{Q} \}$. Must there be some $h \in \mathbb{Q}[x]$ such that $f(x) = g(h(x))$ for all ...

**3**

votes

**1**answer

158 views

### Constructive Proof to Show that Algebraic Numbers are Algebraically Closed

EDIT2: After reading some papers, I think the question can best be rephrased as "How can the minimal polynomial for a polynomial with algebraic coefficients be calculated. I have seen papers and ...

**8**

votes

**2**answers

152 views

### Polynomial-time algorithm for determining whether a polynomial is positive on $\mathbb{N}$

Does there exist a polynomial-time algorithm to determine whether a given polynomial $p(n)$ with integer coefficients is positive on $\mathbb{N}$, in the sense that $p(n) \geq 0$ for all ...

**1**

vote

**0**answers

57 views

### vector space of ternary forms with real rooted property

Let $V \subseteq \mathbb{R}[x,y]_d$ be a two dimensional linear subspace of the vector space of bivariate forms of degree $d$. For each degree $d$ we can find such subspaces with the property that ...

**11**

votes

**1**answer

176 views

### $\pm1$-polynomials with a maximal non-real root

For given $n$, consider a polynomial $\sum_{k=0}^na_kz^k$ with all coefficients $a_k\in\{\pm1\}$. I am interested in the following:
How big can the modulus of a non-real root of such a ...

**2**

votes

**3**answers

109 views

### Determining Roots of a Polynomial with Interval Estimates of Coefficients

Let $f$ be a monic univariate polynomial with real coefficients:
$$f_A(x) = x^n + a_{n-1}x^{n-1} + ... + a_{0}$$
The values of $A=(a_{n-1},...,a_0)$ are unknown, but are estimated as ...

**4**

votes

**0**answers

106 views

### minimal polynomial for a graph

I wonder if there is any result relating the degree $d$ of the minimal polynomial of a directed finite graph to any of its topological features - such as its diameter, or any other similar 'natural' ...

**7**

votes

**1**answer

140 views

### Can Schwartz-Zippel be formulated for commutative rings instead of fields?

The polynomials which occur in the Schwartz-Zippel lemma could be defined for any commutative ring, yet the lemma is restricted to fields. This makes it inapplicable for ...

**1**

vote

**0**answers

34 views

### Request for reference about bound on zeroes of the Laguerre polynomials

Consider the sequence of polynomials given as, $p^{a}_k (x) = (1 - a \frac{d}{dx})^k x^n $ for some parameter $a>0$ and $k$ being a positive integer. For any positive integer $d$ it seems to be ...

**0**

votes

**0**answers

92 views

### Lower bound on number of smooth values of polynomial at primes

Given a polynomial $f$, it is known believed that the number of smooth values of $f$ has a positive proportion (for fixed $u$, $\lim_{X\rightarrow\infty} \frac{|\{ n < X\ :\ f(n)\ is\ X^u\ smooth ...

**2**

votes

**2**answers

77 views

### About the roots of the matching polynomial

Can someone kindly give me an expository reference on matching polynomial and its roots? (there is a proof that they are always real?)
I saw these two related discussions,
Roots of matching ...

**5**

votes

**1**answer

524 views

### Is the set of certain polynomials finite or infinite?

Let us consider the set of all polynomials with the following properties:
i) all coefficients are integer;
ii) the leading coefficient equals one;
iii) all zeros are real and simple and belonging ...

**9**

votes

**3**answers

391 views

### How many isolated roots can a polynomial in $z$ and $\overline{z}$ have?

This is probably well-known by I can't find it now online. My guess is that if the degree is $n$ then it's $2n$ but it's just a hunch.
EDIT:
This is an edited version. Before I asked about roots ...

**2**

votes

**0**answers

52 views

### Behavior of elementary symmetric polynomials near zero sets

It is straightforward to show (see Characterizing intersection of zero sets of elementary symmetric polynomials on R^n) that the set of points $\Lambda_{k}$ in $x \in \mathbb{R}^{n}$ with ...

**3**

votes

**1**answer

43 views

### Polynomial (non-)embedding of a simplex in euclidean space

Let $\Delta$ be a standard $k$-simplex, and let $f:\Delta\to\mathbb R^N$ be a polynomial map with known numerical coefficients. What sort of practical computational algorithms can be used to ...

**2**

votes

**0**answers

134 views

### What are the minimal degrees of the real and imaginary part of an algebraic complex number? [closed]

Let $z=a+bi\in\mathbb C$ with $b\ne0$ be an algebraic complex number of minimal degree $n$. It is obvious that $a=\dfrac {z+\bar{z}}2$ and $b=\dfrac {z-\bar{z}}{2i}$ are also algebraic. For $n=3$, it ...

**2**

votes

**0**answers

91 views

### Uniqueness of the maximum derivative of a rational function

This may seem like an elementary question, but bear with me; you'll find that it is actually quite hard. Consider the function
$$f(x)=\frac{a_nx^n}{\sum_{i=0}^n a_ix^i}$$
with all $a_i\geq 0$ and ...

**3**

votes

**1**answer

125 views

### Real points of zero-dimensional real algebraic varieties

There have been a number of discussions of zeros of random polynomials here (the most recent being: Why do roots of polynomials tend to have absolute value close to 1?).
Here is a closely related ...

**1**

vote

**1**answer

69 views

### Integrals involving trigonometric functions and polynomes

Let $P(x)$ be a real polynome. Specify all such $P(x)$ that one of the next integrals converge:
$$
\int_0^{\infty} sin(P(x))dx, \int_0^{\infty} cos(P(x))dx ?
$$
Among special cases are such ...

**3**

votes

**1**answer

164 views

### A number array related to colored necklaces and the primes

I stumbled upon entry OEIS-A208535 on the enumeration of certain kinds of colored necklaces and noticed that the integers for the odd prime rows of the table there seem to be given by the Moreau ...

**1**

vote

**1**answer

321 views

### A question on polynomials

Let $D=\{0,1\}$.
Are there infinite values of $n\in \Bbb N$ for which there is a polynomial $p(x_1,x_2,\dots,x_n)\in \Bbb R[x_1,x_2,\dots,x_n]$ and a rational function $r(x_1,x_2,\dots,x_n)\in \Bbb ...

**0**

votes

**1**answer

150 views

### Discriminant of a polynomial in two variables

I want to compute the discriminant of the following polynomial
$$
F(X,Y)=X^mY^n+\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}c_{ij}X^iY^j.
$$
Here the discriminate means the equation $D(c_{i,j})$ in the variables ...

**2**

votes

**1**answer

143 views

### Rational functions and polynomials evaluated on a set of points

Let $S$ be a collection of points on the real line.
Let $\{x_i\}_{i=1}^n$ take values in $S$.
Consider a polynomial $p(x_1,x_2,\dots,x_n)$ over $\mathbb R[x_1,x_2,\dots,x_n]$ of degree $d$ which ...

**6**

votes

**1**answer

120 views

### For a Sum-of-Squares cost functions J(x) is it true that J(x)-j* is also SOS?

For polynomial optimization problems the sum-of-squares theory and Lasserre relaxation hierarchy provides a theoretically handy way of getting the solution. There are also results saying that finite ...

**0**

votes

**1**answer

70 views

### About the maximum degree of multivariate polynomial interpolation

It is well known that in the univariate case, to interpolate $k$ points in $\mathbb{R}$, we need to use a polynomial of degree $k-1$.
My question is about multivariate polynomial interpolation in ...

**1**

vote

**0**answers

110 views

### Rings of algebraic integers as quotients of polynomial rings

The ring of integers $\mathcal{O}_K$ of a number field $K$ is always isomorphic to some ring of the form $\mathbb{Z}[x_1, ..., x_r]/\mathfrak{p}$, where $\mathfrak{p} \subset \mathbb{Z}[x_1, ..., ...

**4**

votes

**1**answer

356 views

### Why are the angular differences of these random complex polynomial coefficients almost constant?

This is based on მამუკა ჯიბლაძე's (not-)answer here. I guess it is better to make up a new thread for it.
Let me repeat the setup here: We consider polynomials whose complex roots are randomly ...

**7**

votes

**0**answers

180 views

### Reciprocal polynomials with roots off the unit circle

A polynomial is called reciprocal if its coefficients read forwards are the same as those read backward - there is an obvious translation of reciprocal polynomials into cosine polynomials (since a ...

**3**

votes

**1**answer

67 views

### Multivariate polynomial interpolations

I have a multi-variate, continuous function from $R^n$ to $R$, which I can query for its output for any input. I would like to create an interpolation of that function by sampling a subset of the ...

**0**

votes

**0**answers

69 views

### Algorithm for finding irreducible polynomials in finite field extensions

Let $K(\alpha_1,\ldots,\alpha_n)/K=\tilde{K}/K$ be a finite field extension and suppose we know $\text{irr}(\alpha_1,K)(x),\ldots,\text{irr}(\alpha_n,K)(x)\in K[x]$. Suppose also that we have a basis ...

**213**

votes

**15**answers

30k views

### Why do roots of polynomials tend to have absolute value close to 1?

While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a ...

**2**

votes

**1**answer

42 views

### Bound the degree of the generator of polynomial ring

Suppose we are given two polynomial rings $R_1$ and $R_2$ by presenting their generators, $S_1$ and $S_2$, where $S_i$ are finite set of $m_i$ variables, $i.e.$, $S_i\subset ...

**48**

votes

**2**answers

1k views

### Does one real radical root imply they all are?

Is there an example of an irreducible polynomial $f(x) \in \mathbb{Q}[x]$ with a real root expressible in terms of real radicals and another real root not expressible in terms of real radicals?

**5**

votes

**1**answer

137 views

### Modular polynomials for elliptic curves point counting

The Schoof-Elkies-Atkin (SEA) algorithm (for counting points on elliptic curves over a finite field) performs computations over polynomials modulo some modular polynomials. Originally the "classical" ...