Tagged Questions
2
votes
0answers
64 views
When is an intersection of varieties finite
Consider the general Bezout's theorem: If $p_1 \ldots p_n$ are polynomials with degrees $d_1,\ldots, d_n$ in $\mathbb{R}[x_1,\ldots,x_n]$, with $V = \{a=(a_1,\ldots, a_n) | p_i(a) = 0, \forall i\}$ ...
4
votes
1answer
48 views
A zero for a homogeneous polynomial is a zero for the associated inhomogeneous polynomial
I am trying to prove a simple statement from Reid, Undergraduate Algebraic Geometry, pg 16.
Let $F(U,V)$ be a nonzero homogeneous polynomial of degree $d$:
...
3
votes
1answer
34 views
square system of polynomial equations having infinite number of solutions
Suppose we have a system of $n$ polynomial equations in $n$ unknowns over $\mathbb{C}$ and suppose that the corresponding ideal generated by these equations is not the unit ideal $(1)$. Under what ...
1
vote
0answers
28 views
Homogeneity of translated polynomial
I am currently trying to understand the very basics of complex algebraic curves and I came across the following statement in the book by F. Kirwan (Definition 2.9):
The multiplicity of the complex ...
6
votes
1answer
97 views
What is the correspondence between combinatorial problems and the location of the zeroes of polynomials called?
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
"... is the ...
13
votes
2answers
146 views
Is a linear combination of minors irreducible?
Let $X=(X_{ij})_{1\le i,j\le n}$ be a matrix of indeterminates over $\mathbb C$. For choices $I,J\subseteq\{1,\ldots,n\}$ with $|I|=|J|=k$ denote by $X_{I\times J}$ the matrix $(X_{ij})_{i\in I,j\in ...
1
vote
1answer
107 views
Property of strictly convex polynomial
I have some difficulties in the following problem.
Thank you for all comments and helping.
Let $f:\mathbb{R}^n\rightarrow \mathbb{R} (n\in \mathbb{N})$ be a polynomial.
Suppose that $f$ is strictly ...
2
votes
3answers
67 views
Affine variety over a field
Suppose we have an algebraically closed field $K$. An affine variety is the common zero locus of a collection of polynomials $f_{\alpha} \in K[z_1, \dots, z_n]$. So basically it is the set of points ...
2
votes
1answer
107 views
Over $\mathbb{R}$, if $Z(p') \subset Z(p)$ when does $p' \vert p$?
I'm mainly wondering about the planar case, when $p', p \in \mathbb{R}[x,y]$. For instance the simplest case would be when $Z(p')$ is a line contained in $Z(p)$, does it follow that $p' \vert p$? I ...
3
votes
2answers
144 views
Points and maximal ideals in polynomial rings
Let $k$ be a field, then I want to prove the following statement: for every $P=(b_1,\ldots,b_n)\in K^n$, the ideal $\mathfrak{m}_P=(x_1-b_1,\ldots,x_n-b_n)$ is maximal in the polynomial ring ...
0
votes
1answer
59 views
Equation with 2 variables tricky problem
So let's say I have some random equation $6yx^3 - 3yx + 5 = 0$, but could be anything.
How would I go about finding a value for $y$, that makes it so that this equation only holds true if $x$ is ...
30
votes
7answers
573 views
Appearance of Formal Derivative in Algebra
When studying polynomials, I know it is useful to introduce the concept of a formal derivative. For example, over a field, a polynomial has no repeated roots iff it and its formal derivative are ...
1
vote
0answers
44 views
Positive Semidefiniteness of a polynomial.
I have a multivariate polynomial $p(x_1,\ldots,x_n)$ and I wish to check if it is positive semidefinite over $R^n$. I can always choose any direction $\vec{v}$ and check that the univariate polynomial ...
1
vote
0answers
49 views
Minimal syzygies for polynomial ideals
Let $I$ be an ideal of $S=k[x_1,\dots,x_n]$.
I am asked to find a minimal free graded resolution of $I$, by means of syzygy matrices. I suppose there has to be an algoritmic approach to it, provided ...
0
votes
1answer
225 views
Converting parametric equation to implicit form
So I have the equation defined in homogeneous coordinates $[w; x, y]$ as $[1+t^2; 1-t^2, 2t]$
$$w = 1+t^2$$
$$x = 1-t^2$$
$$y = 2t$$
If I do $w+x-y$ I get $-2t+2$, so $t = -(w+x-y-2)/2$. I was then ...
3
votes
1answer
96 views
Is there an irreducible polynomial vanishing on two components? (In the Zariski sense)
The polynomial
$$f(x,y) = (x^2 − 1)^2 + (y^2 − 1)^2$$
is an example of an irreducible polynomial in $\mathbf{R}[x,y]$ which is irreducible but whose zero set has multiple components in the Zariski ...
1
vote
2answers
82 views
proving that this ideal is radical or the generator is irreducible
How can I prove that the ideal $ (xy-1) \subset k[x,y] $ is radical? I think that it's enough to prove that the polynomial $xy-1$ is irreducible. How can we prove that?
Here $k$ is a field, I'm not ...
4
votes
0answers
78 views
Some elementary facts
What is the simplest and the most conceptual proof of some basic facts on algebraic geometry?
1) Hilbert's Nullstellensatz
2) Regular functions on projective variety - only constants
3) elemination ...
3
votes
1answer
142 views
On the Jacobian Conjecture
I have been asked to do a work on the Jacobian Conjecture for my master's course. While I am familirized with that conjecture and I understand its implications, I would like to ask you all if there is ...
6
votes
1answer
114 views
Solving a polynomial equation with a parametrization
Let $f(x,y)$ be an irreducible polynomial, in the two variables $x$ and $y$.
It sometimes happens that a “lucky” change of variables $x=g(t)$, where $g$ is a non constant polynomial, transforms our ...
0
votes
1answer
40 views
What is the zero set of an exponential polynomial on a torus
Let $a,b,c\in\mathbb C$, and define $$f(x,y)=ae^{i(x+y)}-b(e^{ix}+e^{iy})+c$$ for $x,y\in[-\pi,\pi]$.
For a "generic" triple $a,b,c$ the set $\{f(x,y)=0\}$ consists of two points, but occasionally ...
3
votes
1answer
74 views
Diophantine equations and Groebner bases
I'm trying to teach myself the basics of algebraic geometry and have run into something that I don't understand.
I know that the problem of deciding whether a Diophantine equation $P(\vec{x}) = 0$ ...
4
votes
1answer
109 views
Is $Z(x^2-y^3)$ isomorphic to $Z(y^2-x^3-x^2)$ over the complex numbers?
I'm having trouble determining if the algebraic sets $Z(x^2-y^3)\subset \mathbb{A}^2$ and $Z(y^2-x^3-x^2)\subset\mathbb{A}^2$ are isomorphic over $\mathbb{C}$. My guess is that this boils down to ...
3
votes
2answers
187 views
Injective map on coordinate ring implies surjective?
Suppose that $f:X\rightarrow Y$ is a morphism between two affine varieties over an algebraically closed field $K$.
I believe that if the corresponding morphism of $K-$algebras, ...
2
votes
1answer
49 views
Existence of a non-variety with special properties
Does there exist an infinite set $X\subseteq\mathbb{R}^n$ such that every non-zero polynomial $P\in \mathbb{R}[x_1,x_2,...,x_n]$ has finitely many zeros in $X$?
1
vote
1answer
103 views
Can a polynomial not be in the Reals or Complex?
I was reading Undergraduate Algebraic Geometry by Miles Reid.
and on page 1. It says
If $k$ is in $\textbf{R}$ or $\textbf{C}$ (which it quite often is).
Now I should state some previous ...
0
votes
0answers
86 views
A system of polynomial-like equations
Let $p\in \left[0,1\right]$ and take $a_1,a_2,\ldots,a_n\in \mathbb{R}^{+}$. What is the maximum number of solutions that the system of (nonlinear) equations $$x_1^p +x_2^p +\cdots+x_n^p = 1\\ ...
3
votes
1answer
119 views
Roots of rational equation with multiple variables?
Let's say we have a rational polynomial in $k$ variables. We are only interested in rational solutions. If $k = 1$, name the variables ${x}$, if $k = 2$, name them ${x,y}$.
For $k = 1$, it can be ...
3
votes
1answer
66 views
a question on deforming a ring
I'm trying to learn about deformation theory.
Consider $k[x,y]/\left< y^2 - x^2\right>$.
To deform $k[x,y]/\left< y^2 - x^2\right>$ to make it look like $k[x,y]/\left< ...
1
vote
2answers
181 views
What is Hilbert polynomial of this projective variety?
Suppose you have a map $\varphi\colon\mathbb{C}^m\times\mathbb{C}^n\to\mathrm{Mat}_{m,n}(\mathbb{C})$ defined by sending $(\mathbf{u},\mathbf{v})\mapsto\mathbf{u}\cdot\mathbf{v}^T=(u_i,v_j)$. So ...
5
votes
2answers
217 views
(Minimal?) Polynomials using the Nullstellensatz
I'm struggling with an exercise that was asked in class:
Let $\alpha = \sqrt[3]{3} + \sqrt{7}\sqrt[4]{2}.$ Show that there is a polynomial $p$ in
the ideal $I=\left<a^3 - 3, b^2 - 7, c^4-2, ...
4
votes
1answer
153 views
automorphism of $k[x,y]$ that fix k.
I was doing an exercise, and to finish it, I need to find something, and I conclude that is just this group of automorphism. The questions is, I can say more about it? ( There has some form, etc..) ?
...
1
vote
0answers
47 views
Software for checking whether there is a monomial ordering satisfying some constraints
Monomial orderings (sometimes also called admissible orderings and term orderings) play a crucial on the topic of Gröbner bases, and so
there is a huge literature on them. They are essentially linear ...
1
vote
1answer
135 views
An irreducible polynomial over $\mathbb{R}[x,y]$
Let $(x_i,y_i)$ be a finite subset of points of $\mathbb{R}^2$. Find an irreducible polynomials $f(x,y)$ over $\mathbb{R}[x,y]$ such that vanish only in that points.
EDITED: Where $\mathbb{R}$ ...
4
votes
1answer
305 views
Zariski Open Set
Consider the Zariski Topology on $\mathbb{C}^n.$ Then is it true that for every non-empty Zariski open set $U,$ $U \cap \mathbb{R}^n$ is open dense in $\mathbb{R}^n$?
0
votes
1answer
78 views
Solution Set of a Polynomial System
Consider the polynomial System $F(x)-c=0,$ where $F:\mathbb{C}^n \rightarrow \mathbb{C}^n.$ Is it true that for almost all values of $c\in \mathbb{C}^n,$ the polynomial system will only have isolated ...
0
votes
1answer
90 views
Show that (vector) subspaces of $\mathbb{A}^n$ are algebraic sets
i have just started to learn some algebraic geometry and there is a statement in the notes i am following that i do not understand: "Subvector spaces of $\mathbb{A}^n$ are algebraic sets. They are of ...
1
vote
0answers
92 views
Complexity of finding solutions for a system of polynomial equations
Problem A: Given a set of polynomial equations $ f_1=0,\ldots,f_m=0 $, where the $ f_i $'s are multivariate polynomials with $ n $ variables over a field $\mathbb F$, decide whether there is a ...
1
vote
1answer
131 views
Computing the power of real algebraic numbers
I'm looking for an efficient algorithm to compute the $n$-th power $\alpha^n$ of a real algebraic number $\alpha$ given by an interval representation for $n \in \mathbb{N}$. An interval representation ...
1
vote
0answers
41 views
Small generating set of third degree polynomials in $R=\mathbb{Z}_2[x_1,\dots,x_n]/\langle x_1^2-x_1,\dots,x_n^2-x\rangle$
Let $R=\mathbb{Z}_2[x_1,\dots,x_n]/\langle x_1^2-x_1,\dots,x_n^2-x\rangle$, i.e. we can think of $R$ as the ring of multivariate polynomials with the additional property that one can "linearize" ...
1
vote
1answer
83 views
Fitting a quadratic polynomial of a special form
Consider a real polynomial
$$ f(x) := \sum_{k=0}^n c_k x^k $$
of degree $n$, where $c_k \in \mathbb{R}$ are constant coefficients. Given $n+1$ samples $f_i$ of $f$ at distinct sample points $x_i$, ...
7
votes
2answers
516 views
Is there a General Formula for the Transition Matrix from Products of Elementary Symmetric Polynomials to Monomial Symmetric Functions?
Given the elementary symmetric polynomials $e_k(X_1,X_2,...,X_N)$ generated via
$$
\prod_{k=1}^{N} (t+X_k) = e_0t^N + e_1t^{N-1} + \cdots + e_N.
$$
How can one get the monomial symmetric functions ...
7
votes
0answers
141 views
What is the dimension of this algebraic variety?
Let $\mathbb K$ be a number field of degree $n$ over $\mathbb Q$, and let
$\alpha_1,\alpha_2, \ldots ,\alpha_n$ be a $\mathbb Q$-basis of $\mathbb K$. Then there
are coefficients $(c^{ij}_k)$ (where ...
0
votes
1answer
66 views
Natural space to consider solution to polynomial equations
Why is the complex projective planes the most natural place to look to consider solutions of polynomial equation?
Why is the complex plane $\mathbb{C}$ adequate for polynomial equations of one ...
0
votes
1answer
71 views
Check for algebraic function?
How do I check whether a give function is 'algebraic' or not? I have a function $m(z) = 2\pi i z^n$ where $z \in \mathbb{C} \backslash \mathbb{R}$ and $n \in \mathbb{Z}$. I can write this as $w - 2\pi ...
2
votes
1answer
179 views
Comparing algebraic varieties over a shared subset of variables
I'm currently experimenting with polynomial ideals and Gröbner bases, and I seem to be lacking some terminology/understanding.
I have two systems of polynomial equations $P$ and $Q$ over a field ...
3
votes
0answers
117 views
General questions about level sets of polynomials of two variables
Sorry if I'm being too general here, but here it goes. I'm trying to find out more about levels sets of polynomials of two variables of degree $d$
$$ C = \{ (x,y) \ : \sum_{1 \leq i + j \leq d} ...
7
votes
1answer
498 views
(Ir)reducibility criteria for homogeneous polynomials
Suppose I have a homogeneous polynomial in at least 3 variables over some algebraically closed field (of characteristic 0, if need be). Question: How may I test — by hand — whether it is irreducible?
...
7
votes
1answer
162 views
From complex solution to solutions over finite fields
There are several ways (Hilbert's Nullstellensatz, model theory, transcendence bases etc.) to prove the following (amazing!) result:
If $f_1,...,f_r$ is a system of polynomials in $n$ variables with ...
10
votes
3answers
491 views
Why is there no polynomial parametrization for the circle?
How does one show that the unit circle admits no polynomial parametrization?
What is needed for this, are there general criteria?
Thanks
