# Tagged Questions

**3**

votes

**0**answers

60 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 ...

**0**

votes

**1**answer

129 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 ...

**0**

votes

**0**answers

104 views

### Composition of multilinear forms agreeing on a subset of points

Let $n$ be a perfect square.
Consider multiaffine polynomials $p(x_1,x_2,\dots,x_n),q(x_1,x_2,\dots,x_n),r(x_1,x_2,\dots,x_n),$$\{s_j(x_1,x_2,\dots,x_n)\}_{j=1}^{n}$$\in\mathbb R[x_1,x_2,\dots,x_n]$.
...

**6**

votes

**1**answer

115 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 ...

**5**

votes

**1**answer

120 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" ...

**10**

votes

**1**answer

561 views

### Ways to show a system of polynomial equations has no solution

I came across the following system of polynomial equations on $X_1,\dots,X_{m-2}$:
$$
\begin{cases}
2X_{2s}+\sum\limits_{t=1}^{2s-1}(-1)^tX_tX_{2s-t}=0,\quad s=1,\dots,\frac{m}{2}-1,\\
...

**0**

votes

**1**answer

194 views

### Decomposition of symmetric homogeneous polynomials

Can every symmetric polynomial of degree $r$ in $d$ variables that has no constant term be written as a sum of the $r$th powers of linear polynomials in $d$ variables and a homogeneous polynomial of ...

**3**

votes

**0**answers

118 views

### Bound for the height of equations defining the singular locus of a variety

Fix positive integers $m, n, d$.
In what follows, the height of an algebraic number will mean the absolute multiplicative height.
Let $V \subset \bar{\mathbb{Q}}^n$ be an affine algebraic variety ...

**1**

vote

**0**answers

107 views

### First Description of how to Remove Radicals from Equations

Who first described the technique of removing radicals as indicated in the answers to questions Tools for Removing Radicals from Equations and Rewrite sum of radicals equation as polynomial equation ? ...

**1**

vote

**0**answers

246 views

### A question on partial fraction decompositions

This question concerns a mapping from the poles of a rational function to its partial fraction decomposition coefficients. We assume that the rational function is the inverse of a polynomial of degree ...

**4**

votes

**1**answer

175 views

### Stronger versions of Schwartz-Zippel for random linear subspaces

This is a (self-contained) followup question to http://math.stackexchange.com/questions/380672/analogue-of-the-schwartz-zippel-lemma-for-subspaces.
Let $f : \mathbb{R}^n \to \mathbb{R}$ be a nonzero ...

**3**

votes

**2**answers

326 views

### Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$

I’m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results ...

**3**

votes

**0**answers

120 views

### Question related to the number of rational curves on a hypersurface

The following is from a PhD thesis "Algebraic Circle Method" by Thibaut Pugin, which I am currently reading.
Let $k$ be a finite field of order $q$.
Let $\mathfrak{X} \subseteq \mathbb{P}^n_k$ be ...

**1**

vote

**1**answer

155 views

### Zeroes of a complex polynomial on a sphere as a manifold

Let $ f \in \mathbb{C}[z_1, \ldots, z_n]$ be a polynomial such that $f'(z) \neq 0$ if $z \neq 0$ ($f'$ means $\left( \frac{\partial f}{\partial z_1}, \ldots, \frac{\partial f}{\partial z_n}\right)$ ). ...

**3**

votes

**0**answers

128 views

### Number of rational curves on varieties over finite fields

Let $k$ be a finite field. Let $P_r$ be set of degree $r$ binary forms
over $k$.
We define
$$
\mathcal{M}_r = \{ (x_0, ..., x_n) \in P_r^{n+1} : x_0^d + ... + x_n^d = 0 \text{ and the }x_i \text{'s ...

**6**

votes

**0**answers

59 views

### Bounding volume of cell in complement of zero set

I am given an integer polynomial $f \in \mathbb{Z}[X_1, \ldots, X_n]$ of bounded degree and bounded coefficient size. The polynomial's zero set partitions $\mathbb{R}^n$ into cells. What I am looking ...

**5**

votes

**2**answers

269 views

### Can we decompose a polynomial into difference of convex polynomials?

Given a multivariate polynomial $p(x_1, ..., x_n)$ on $\mathbb{R}^n$, can we always decompose it into the difference of two convex polynomials? i.e., is there a pair of convex polynomials $f$ and $g$, ...

**0**

votes

**1**answer

146 views

### Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial

The following question is motivated by the study of a stability border for a robust linear time-invariant control system.
Let us we have an affine family of $n\times n$ matrices with indeterminate ...

**2**

votes

**1**answer

204 views

### Powers of linear functions span the space of polynomial functions?

Let $P_n$ be the space of polynomials in $n$ variables over a field of characteristic 0.
I'm very sure that the space spanned by powers of linear functions is the whole space $P_n$.
Anyone can come ...

**3**

votes

**1**answer

203 views

### Are there any efficient (polynomial time) algorithms for finding if a multivariate quadratic polynomial has a root?

I know that in general, polynomial satisfiability is NP; however, I'm curious to know what work has been done on special classes of polynomials, and in particular quadratic polynomials of multiple ...

**1**

vote

**2**answers

272 views

### Bounds on the largest root of a polynomial

Consider the following polynomial: $p(x)=x^{3}-(k-1)x^{2}-(2k-1)x+(k-1)^{2}$, where $k \geq 5$ is a fixed parameter. I am trying to find a strong lower bound on the largest root $x_{\max}$ of the ...

**0**

votes

**0**answers

137 views

### Image of critical points

Let $K$ be a field of characteristic $0$, $f:K^n\rightarrow K^n$ be an algebraic function, that is, $n$ polynomial functions in $n$ variables. Let $S$ be the set of critical points of $f$. If ...

**14**

votes

**1**answer

582 views

### When is $f(x_1, \dots, x_n)+c$ an irreducible polynomial for almost all constants $c$?

Let $f\in \mathbb C[x_1, \dots, x_n]$, $n\ge 1$, be a non-constant polynomial. Consider the polynomial $f+t\in \mathbb C[t, x_1,\dots, x_n]$. This is an irreducible polynomial in $\mathbb C(t)[x_1, ...

**7**

votes

**0**answers

454 views

### Getting a bound via polynomial equations

When studying the existence problem of power residue deference sets, I came across the following system of polynomial equations over $\mathbb{C}$,
\begin{cases}
...

**2**

votes

**1**answer

323 views

### About the practice of Bernstein-Kushnirenko theorem

The following refers to
common roots of bivariate polynomial equations and, in particular to the quim's and auniket's comments.
The BKK theorem (cf. arXiv:0812.4688. Theorem 5.4) asserts that if we ...

**3**

votes

**0**answers

115 views

### Elliptic curves and quasi-self-reciprocal polynomials

I am reading Shoichi Kihara's On the rank of the elliptic curve $y^2=x^3+k, II$ [Proc. Japan Acad. Ser. A Math. Sci. Volume 72, Number 10 (1996), 228-229] which is available here ...

**5**

votes

**1**answer

331 views

### Is an ideal generated by multilinear, irreducible, homogeneous polynomials of different degrees always radical?

I asked this question on math.se and someone even put a bounty on it, yet there was no answer. Hence, I am asking here. Assume $\Bbbk$ to be a field of characteristic zero.
Definition. A ...

**3**

votes

**0**answers

138 views

### Basis of multivariate polynomials with a specified set of roots

I'm studying certain polynomials of 2 complex variables, say x and y. These polynomials have roots at the non-negative integers, that is both $x$ and $y$ have to be
$x,y \in \mathbb{N}$
...

**7**

votes

**3**answers

374 views

### How few terms may appear in a polynomial with given (cyclotomic) roots and nonnegative coefficients?

Given $W \subset \mathbb C$, let $S_W$ be the set of polynomials in $\mathbb R[x]$ that vanish on $W$ and have only nonnegative coefficients.
Warm-up question: It's clear that if $W$ contains a ...

**1**

vote

**1**answer

315 views

### Is there a quick way to find all roots of a real polynomial with multiple variables?

If I am asked to find the roots of a polynomial of one variable, I will use a computer to estimate the eigenvalues of its companion matrix. Now suppose I'm given a real polynomial of multiple ...

**7**

votes

**2**answers

191 views

### Finding a low-degree polynomial vanishing on half the zeroes of a polynomial system

Let $f(x)$ be a real polynomial of degree $2d$ without real roots. Let the complex roots be $z_1$, $\bar{z_1}$, $z_2$, $\bar{z_2}$, ..., $z_d$, $\bar{z_d}$ with $z_i$ in the upper half plane. Let ...

**4**

votes

**0**answers

163 views

### Pair of two-variable polynomial equations of high order

I have the following pair of equations to be solved for two variables $\rho$ and $D$ resulting from a certain Maximum Likelihood Estimation for a time series $X_n > 0$, $n=0, \ldots, N+1$ with $N ...

**4**

votes

**4**answers

556 views

### The Icosahedron Equation

$$1728 V^5 + F^3 = E^2 \;.$$
Can anyone point me to a concise, modern derivation and explanation of
the significance of the icosahedron equation, more modern and
concise than Klein's description in ...

**1**

vote

**2**answers

322 views

### Existence of non-trivial solution to non linear polynomial system

I need to find conditions for the existence of non-trivial solutions to a multivariable polynomial system in two cases:
The first case:
$f1: a_1x^2+a_2xy+a_3y^2+a_4z^2=0$
$f2: ...

**2**

votes

**0**answers

174 views

### Reducing a System of Polynomial Equations

I am currently writing a program in SAGE which computes Nilpotent Orbit Varieties for an Algebraic Geometry research project and I have reduced my problem to the following:
Consider a system of ...

**7**

votes

**4**answers

517 views

### Minimal representation of a polynomial as a linear combination of squares

Given a polynomial of degree $2n$ over $\mathbb{Q}$, how to represent it as a linear combination (with rational coefficients) of squares of polynomials of degree at most $n$ over $\mathbb{Q}$ such ...

**1**

vote

**1**answer

555 views

### common roots of bivariate polynomial equations

Let $f(x,y)=0$ and $g(x,y)=0$ be bivariate polynomial equations where the polynomials have the same degree, say, $N\geq 3$. Furthermore, both of them have the same terms but different coefficients. ...

**0**

votes

**1**answer

192 views

### Zeros of compositions of polynomials and derivatives

Suppose we have polynomials $f$ and $g$ over a field $F$ and for some $a\in F$ we know $(x-a)^m$ divides $f\circ g$, i.e. $f\circ g$ has a zero of order $m$ at $a$. Then we have that $(x-a)^{m-n}$ ...

**6**

votes

**3**answers

420 views

### What is the correspondence between combinatorial problems and the location of the zeroes of polynomials called?

(From MSE)
In the wikipedia article on the Italian-born American mathematician and philosopher Gian-Carlo Rota, it is stated that the one combinatorial idea he would like to be remembered for
...

**1**

vote

**1**answer

232 views

### Homogenous polynomials as sum or differences of squares and symmetric polynomials

I seem to recall that a general homogenous real polynomial $P$ of even degree in $n$ variables, $n\geq 3,$
cannot always be expressed as $P(x_1,\dotsc,x_n)=\sum_j a_j Q_j^2(x_1,\dotsc,x_n)$ where $a_j ...

**8**

votes

**1**answer

358 views

### Constructive proof of “Projective implies proper”

For every ring $A$, the structural morphism of schemes $\pi_A : {\bf P}^n_{A} \to {\rm Spec}{A}$ is a closed map. The usual proof of this fact is not constructive : given equations of a closed subset ...

**5**

votes

**3**answers

901 views

### Is there any theorem like implicit function theorem in $\mathbb{Q}$ ?

My qeustion is that,
is there any theorem like implicit function theorem in $\mathbb{Q}$ ?
More precisely, let $p(\bar{x},\bar{y})$ be in $\mathbb{Z}[\bar{x},\bar{y}]$ such that in $\mathbb{Q}$, for ...

**2**

votes

**0**answers

288 views

### Solving 3D equation system (inverse-projecting a triangle)

Please, how is the equation system below named exactly (to search further literature)?
Does it have an analytical solution? If it doesn't, then what could be the fastest numerical method for it ...

**12**

votes

**4**answers

536 views

### Partitions of $\mathbb{R}^d$ by implicit polynomial equations

Given a polynomial
$p(x_1,x_2,\ldots,x_d)$
in $d$ variables, with maximum degree $k$,
what is the maximum number of
components of $\mathbb{R}^d$ minus $p(\ldots)=0$?
In other words, into how many ...

**16**

votes

**2**answers

1k views

### When is P(x)-Q(y) irreducible?

Let $k$ be an algebraically closed field (in my application, it is characteristic zero, but this probably doesn't matter so much), and let $P: k \to k$, $Q: k \to k$ be polynomials of one variable. ...

**5**

votes

**1**answer

218 views

### Convexity of a specific semialgebraic set

I have an engineering problem which maybe resolved with semi-definite programming optimization.
I have a set which I would like to know if is convex:
Being $m \in \mathbb{R}^+$ a positive real ...

**7**

votes

**1**answer

550 views

### distinct zero points for polynomial

I met an interesting phenomenon. Suppose $f(z)=\frac{1}{p(z)}$ where p(z) is a polynomial in $\mathbb{C}[z] $. If there exists a $ k \in \mathbb{N} $ and $ k>1 $ such that after you take $k$-th ...

**11**

votes

**2**answers

436 views

### Can any numerical polynomial be a Hilbert polynomial of something?

is it true that any numerical polynomial , i.e. $f(t)\in \mathbb Q[t], f(n)\in\mathbb Z\ \forall n\in\mathbb Z\ $ can be presented as Hilbert polynomial of some variety?

**2**

votes

**1**answer

216 views

### Are there polynomials (almost) all of whose intersection numbers are divisible by some integer?

I've been playing around with some basic intersection theory, and I've wondered the following:
For every two integers $n$ and $m$, and complex numbers $a_1,...,a_n$, are there polynomials ...

**3**

votes

**1**answer

288 views

### Are there n polynomials for which all intersection multiplicities are at least m?

I don't know whether this is known or not, but I was thinking of the following problem.
Let $n$ and $m$ be natural numbers. Are there $n$ polynomial $f_1,...,f_n\in \mathbb{C}[x]$, such that all of ...