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.
2,555
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Intersection Solutions of four nonlinear equations
I have four nonlinear equations I want to find the points of intersection of these equations, and I used the software Mathematica, unfortunately after many hours of waiting it does not give me any ...
0
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1
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405
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Reason Coppersmith fails here?
Take classic problem of finding $P,Q$ in balanced semi-prime $N=PQ$.
$P$ has a binary expansion and so does $Q$. We can set the binary $0/1$ variables to be $x_1$ through $x_{\lceil\log P\rceil}$ and $...
9
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1
answer
295
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Nonnegative coefficients of a product of polynomials
Let $P(x_1,\dots,x_n)\in\mathbb{R}[x_1,\dots,x_n]$. Does there exist an algorithm to decide whether there is a nonzero polynomial $Q(x_1,\dots,x_n)\in\mathbb{R}[x_1,\dots,x_n]$ such that the product $...
3
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1
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181
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Divergence of a series related to Schinzel's hypothesis H
The Series
Consider the series identity
$$\Phi(s) = \sum_{n=1}^\infty \frac{\mu(n) (\log n)^k}{n^s} \sum_{r \in R(n)} \zeta(s,r/n) = \sum_{n=1}^\infty \frac{\Lambda_k'(f(n))}{n^s}$$
$$R(n) = \left\...
8
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0
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351
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How localized can a polynomial be in the L1 norm?
Let $0<s<2$ be a parameter, $\Omega = [-1,1]$, and $\Omega_s\subset \Omega$ be a set of measure $s$. I would like to bound the following ratio from above:
$$\sup_{p\in\mathcal{P}_n} \frac{\...
2
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1
answer
166
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On a special type of subring of $\mathbb C[x_0,...,x_{q-1}]$
Let $p,q$ be odd primes. Consider the polynomial ring $\mathbb C[x_0,...,x_{q-1}]$. For $m=0,1,...,p-1$, let
$$\sigma_m=\sum_{0\le j_0\le p;...;0\le j_{q-1}\le p; j_1+...+j_{q-1}=p; 1.j_1+...+(q-1)...
13
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Products of Cyclotomic Polynomials with Nonnegative Coefficients
I'm curious if there are any results that allow us to determine if a product of cyclotomic polynomials (not necessarily all distinct) results in a polynomial having nonnegative coefficients.
Some ...
1
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1
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289
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System of polynomial equations with a known root
I have 5 polynomial equations for 5 variables and I know that the set of roots is finite. All coefficients are integers. Ultimately I'd like to find all roots but finding the Groebner basis is ...
8
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0
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168
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An Ehrhart positivity question related to Schur polynomials
Consider the Schur polynomial $s_\lambda(x_1,\dotsc,x_k)$.
It is easy to see from the hook-content formula for counting the number of semi-standard tableaux, that the function
$$
n \to s_{n \lambda}(1,...
5
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1
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Complex polynomial
Let $N$ be a big integer number and consider the equation :
$$ x^{N} + a_{N-1} x^{N-1} + ...+a_{1} x + o(\frac{1}{N})=0,$$
where $o(h)$ is by definition a term such that $\lim_{h \to 0} o(h)/h =0$. ...
4
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1
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680
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Marsden's Identity and B-splines
Marsden's Identity states that for every $\tau$ in $\mathbb{R }$:
$$(\cdot -\tau)^{k-1}=\sum_j\Psi_{j,k}(\tau)B_{j,k,t} \, ,$$
with $\Psi_{j,k}=(t_j-\tau)\times...\times(t_{j+k-1}-\tau)$.
Following ...
7
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0
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Maximal subalgebras in polynomial ring $\mathbb{R}[x]$ over the field $\mathbb{R}$ of real numbers
Question. What are the maximal subalgebras of polynomial ring $\mathbb{R}[x]$ over the field of real numbers?
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How do you quickly determine which coefficients are greater than zero when multiplying two univariate positive polynomials?
Suppose that I have two polynomials with a degree of $n$, $A(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0$ and $B(x) = b_nx^n + b_{n-1}x^{n-1} + ... + b_0$ and the coefficients of these polynomials are ...
4
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Subsets $E$ of $\mathbb{F}_{p^k}$ with vanishing polynomial subset sums
The following question arose in some discussions recently as a misunderstanding of another problem.
Question: Which subsets $E\subset \mathbb{F}_{p^k}$ satisfy the property that $ \sum\limits_{x\in E}...
2
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1
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Bernstein bound on the number of roots of a rectangular multivariable polynomial systems
I would like to know what Bernstein's bound is on multivariate polynomial systems where there are more equations than unknowns. I have a "generic" zero-dimensional multivariable polynomial system with ...
1
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0
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247
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Method of Coppersmith optimal for multivariate?
It is shown that Coppersmith method yields optimal integer root extraction for univariate polynomials in https://arxiv.org/abs/1605.08065 and a follow up work attempts this for bivariate polynomials ...
3
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0
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Independence of number fields generated by roots of Littlewood polynomials
Let $\mathcal{R}_d \subset \bar{\mathbb{Q}}$ be the set of all roots of degree $d$ polynomials with $\{-1,0,1\}$ coefficients and
$$
c(d) := \min_{\substack{ \alpha, \beta \in \mathcal{R}_d \\ \alpha^...
6
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$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$ is always an integer
Does anyone know if the following problem has ever been studied?
Let $a$ and $b$ be two real numbers and consider the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$$
where $n$ is a ...
1
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0
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About Kazhdan Lusztig polynomial evaluating at q=1
Given $w\le w'$ (in Bruhat ordering), does $P_{x,w}(1)\le P_{x,w'}(1)$ (in usual ordering of $\mathbb{R}$), where $P_{x,w}(q)$ is the Kazhdan Lusztig polynomial?
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Prime generating polynomials
Continuation to this previous question.
According to Lemke-Oliver, an irreducible polynomial $G$ of degree $g$ with positive leading coefficient and $\Gamma_G\neq0$ (with $\Gamma_G$ a certain factor ...
9
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0
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244
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Semi-primes represented by quadratic polynomials
According to Lemke-Oliver, irreducible quadratic polynomials $G$ with positive leading coefficient and $\rho(2)<2$, (where $\rho(m)$ denotes the number of incongruent solutions to the congruence $G(...
3
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143
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Solutions to $w_x=CA_x$, $w_y=CA_y$ other than $w=f(A)$ and $C=f'(A)$?
Let $R \in \{\mathbb{C}[t^2,t^3], \mathbb{Z}[\sqrt{5}]\}$,
and let $A=A(x,y) \in R[x,y]$ with $\deg(A) \geq 1$ (total degree).
I wish to prove or find a counterexample to the following claim:
If ...
3
votes
1
answer
263
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Positive real root separation (v2)
(This is a follow-up question to Positive real root separation)
Let $\beta\in(1,2)$ and $\gamma\in(1,2)$ be Galois conjugates of height 1. That is, there exists a polynomial $p$ with coefficients $-1,...
9
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0
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326
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Is this a possible strengthening of the Lehmer conjecture?
Here is another possible refinement of the Lehmer conjecture.
For $\alpha \in \overline{\mathbb{Q}}^{\times}$, let $C_{\alpha} \subseteq \mathbb{Q}(\alpha)$ be the maximal cyclotomic field contained ...
4
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1
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253
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Positive real root separation
Let $\beta\in(1,2)$ and $\gamma\in(1,2)$ be Galois conjugates of height 1. That is, there exists a polynomial $p$ with coefficients $-1,0,1$ such that $p(\beta)=p(\gamma)=0$ (not necessarily minimal)....
2
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0
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Asymptotics of Littlewood polynomials
Littlewood in [L] states several conjectures regarding asymptotics of polynomials with $\pm1$ coefficients.
He considers the class $\mathscr F$ of polynomials of form $\sum^n\pm z^m$ and asks whether ...
3
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1
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317
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Roots of anti-palindromic polynomial if coefficients are odd.
This is in continuation of the question asked in this earlier post here. Given an anti palindromic polynomial of degree $n$ with odd coefficients, does it have roots on the unit circle?
6
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$f,g \in \mathbb{Z}[x,y]$ satisfying: $\operatorname{Jac}(f,g)=0$ and $f,g \notin \mathbb{Z}[h]$ for every $h \in \mathbb{Z}[x,y]$?
Is it possible to find $f,g \in \mathbb{Z}[x,y]$ (with $\deg(f),\deg(g) \geq 1$) such that the following two conditions are satisfied:
(1) $\operatorname{Jac}(f,g)=f_xg_y-f_yg_x = 0$.
(2) ...
1
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1
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Reference request for anti-palindromic polynomials.
I have come across a lot of papers that are written about the palindromic polynomials, however, I am recently interested in polynomials satisfying
$$f(-x) = x^nf(1/x)$$
for $n\geq 1$ and for all $x\...
2
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1
answer
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For every table of interpolating nodes, there is a positive continuous function whose interpolating polynomials are not positive infinitely often
Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that ...
4
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1
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292
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A Conjecture about the integral related to Chebyshev polynomial
I am interested in the following integral related to the Chebyshev polynomials
$$I_{n,m}:= \int_0^\pi \left(\frac {\sin nx}{\sin x}\right)^{m} dx,$$
where $n,m\in \mathbb{Z}^+.$
It is easy to see ...
0
votes
1
answer
167
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For which $n$, can we find a sequence of $n+1$ distinct points s.t. the interpolating polynomial of every +ve continuous function is itself +ve
Fix an interval $[a,b]$. For which integers $n>1$, does there exist $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that for every continuous function $f:[a,b] \to (0,\infty)$, the ...
4
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1
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992
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Orthogonal basis of polynomials?
Let us define the basis of polynomials given by:
$$
\begin{array}\
P_0=1, \\
P_1=x, \\
P_2=x(x-1), \\
P_3=x(x-1)(x-2), \\
P_4=x(x-1)(x-2)(x-3), \ldots\\
\end{array}
$$
I would like to know if this ...
22
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2
answers
647
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Does every positive continuous function have a non-negative interpolating polynomial of every degree?
Let $f:[a,b] \to (0,\infty)$ be a continuous function. Then is it necessarily true that for every $n\ge 1$, we can find $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that the ...
2
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1
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$(x + y + z)....(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$ To find $P$
$$(x + y + z)(x + y\omega_n + z\omega_n^{n-1})(x + y\omega_n^2 + z\omega_n^{n-2})....(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$$ where $\omega_n$ is an nth root of unity.
The ...
0
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0
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244
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gcd of polynomial values
Suppose that $f$ and $g$ are two coprime polynomials in $\mathbb Z[x]$. I'm interested in any sort of upper bound on $gcd(f(a),g(a))$, in terms of the integer $a$. Are there any results of this type?...
3
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1
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133
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A geometric property about certain polynomials in two variables
Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$
where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last ...
4
votes
1
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239
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Is there any Menelaus-type theorem for polynomials?
Consider $n+3$ polynomials of degree $n$, say $P_1(x) , ... P_{n+3}(x)$.
In addition, consider that there are distinct (it is not necessary that all of them be distinct) numbers $x_{ij}$ for $1 \...
7
votes
1
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244
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The determinant of a $4\times4$ matrix associated to some specific polynomial as follow
Let $f\in \mathbb{R}[x_1,x_2,x_3,x_4]$ defined by
$$f_a(x_1,x_2,x_3,x_4)=\prod_{1\leqslant i<j\leqslant4}(x_i-x_j)^{2a_{ij}}$$
where $a=(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})\in \mathbb{N}^6$.
...
0
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1
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99
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Algebraic independence of certain values implies algebraic independence of functions?
It is quite general and elementary question.
Is it possible that some holomorphic functions $f_1,\cdots,f_m $ on a region $\Omega $ of $\mathbb C$ satisfies:
Whenever $(f_1(z), \cdots, f_m (z)) $ is ...
8
votes
0
answers
215
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Is there an approximate formula for the discriminant of a sparse polynomial?
Consider integer polynomials $P \in \mathbb{Z}[X] \setminus \{0\}$ of a degree $D \geq 1$ and without multiple complex roots. Let me introduce a notation
$$
d(P) := \frac{1}{D} \log{|\mathrm{Disc}(P)|}...
4
votes
1
answer
136
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Polynomial expansions via prime-base digits
Fix a prime number $p$. If $n$ is a positive integer, then denote
$$\text{$\omega_{p,k}(n):=\#$ of $k$'s in the $p$-ary expansion of $n$}$$
and the total sum of all its $p$-ary digits by
$$\Omega_p(...
3
votes
1
answer
239
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Solving polynomial inequalities -- efficient Positivstellensatz on a computer
I have about twenty five (multilinear) polynomials $f_1(\mathbf{x}), f_2(\mathbf{x}), \dots, f_{25}(\mathbf{x})$ all in fifteen variables and I would like to decide if there is a $\mathbf{y} \in [0,1]^...
2
votes
0
answers
57
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Classes of curves with "determinant-like operation"
Consider a motivating example:
Let $E\in \mathbb{Q}[y][x]$ be of degree $n=2$ (in $x$) and separable when viewed as a member of $\mathbb{Q}[x,y]$. Therefore we can calculate it's roots in $\mathbb{Q}[...
4
votes
0
answers
114
views
Name of a polynomial basis
Does anybody know if the set of polynomials
$\{ 1,x,x^2+2,x^3+3x,x^4+4x^2+6,{x}^{5}+5\,{x}^{3}+10\,x,{x}^{6}+6\,{x}^{4}+15\,{x}^{2}+20,\ldots \}$
appear in the literature? I am curious what they are ...
4
votes
1
answer
126
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On the linear factors of a polynomial obtained from the determinant of a matrix whose entries are related to Binomial expansion
Consider the polynomial ring $R=\mathbb C[x,y]$.
Consider the matrix $A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&...
2
votes
0
answers
95
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Polynomials passing through points with tangential conditions
In corollary here http://math.mit.edu/~lguth/Exposition/erdossurvey.pdf on polynomial methods it is said "(Parameter counting) If $S\subset\mathbb F^n$ is a finite set, then there is a non-zero ...
9
votes
0
answers
340
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Which polynomials in the minors of a matrix are invariant under conjugation?
$\newcommand{\Cof}{\operatorname{cof}}$
This is a cross-post.
Let $1<k<n$ be a fixed integer. I am trying to understand "what can be said" about the $k$-degree minors of a linear map $T:V \to ...
3
votes
0
answers
110
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Relationship bewteen Kazhdan-Lusztig Vogan polynomial and Kazhdan-Lusztig polynomial
Let $W^I=\{w\in W: w^{-1}\Phi_I^+\subseteq \Phi^+\}$, $W_I$ be the Weyl group generated by $I$ and $w_I$ be the longest element in $W_I$
Let $M(\lambda)$ be the Verma module with highest weight $\...
5
votes
0
answers
146
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Descending chain of subalgebras of $k[x,y]$
Let $k$ be a field of characteristic zero.
Let $\{R_i\}_{i \in \mathbb{N}}$ be a descending chain of $k$-subalgebras of $k[x,y]$: $k[x,y]=:R_0 \supseteq R_1 \supseteq R_2 \supseteq \ldots$,
such that ...