All Questions
17 questions
1
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0
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164
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Combinatorial question related to Hankel-type matrices
Let $\mathbb{N}$ be the set of non-negative integers. Let $n\geq 2, d$ be positive integers. I would like a lower bound on the largest integer $r$ for which the following property holds:
For any ...
- 980
3
votes
2
answers
451
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Guess the next polynoms in the sequence (MO vs. AI :), count anticommuting $F_p$-matrices, P. Hrubeš conjecture
Here is a sequence of polynoms - (presumably) counting N-tuples of ANTI-commuting 2x2 matrices over $F_p, p>2$. (That is just the case of 2x2 matrices, and (surprisingly) it is not so easy to see a ...
- 24.9k
6
votes
1
answer
288
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The combinatorics of the Nullstellensatz for the variety of nilpotent matrices
Let $H_n$ denote the set of $n \times n$ nilpotent matrices with complex entries. The set $H_n$ may be regarded as an algebraic variety. Indeed, consider the polynomial ring $\mathbb{C}[A_{i,j} : 1 \...
- 484
3
votes
0
answers
107
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Non-tree models of Lagrange inversion polynomials
The specific Lagrange inversion / series reversion polynomials (LIPs) I'm addressing are illustrated in OEIS A134685 with a general linear term and in Lang's pdf for A176740 with the coefficient of ...
- 10.5k
5
votes
1
answer
163
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Polynomials vanishing on prescribed layers
Given a prime $p$ and an integer $n\ge p$, what is the smallest possible degree of a polynomial $Q\in\mathbb F_p[x_1,\dotsc, x_n]$ such that $Q$ vanishes on every vector $x\in\{0,1\}^n$ of weight $w(x)...
- 23k
2
votes
0
answers
112
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Getzler's stable graphs for modular operads
In The semi-classical approximation for modular operads, Getzler displays a table at the bottom of page two enumerating certain stable graphs. (This is related to the MO-Q "Stable graphs: Feynman ...
- 10.5k
1
vote
1
answer
241
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Integral zeros of a multivariate polynomial
Consider the multivariate polynomial
$$f(x_1,\ldots,x_m)=mk\sum_{i=1}^mx_i^2-mk(k-1)\sum_{i=1}^mx_i-\left(\sum_{i=1}^mx_i\right)^2,$$
for integers $m,k\ge2$. We are looking for integral zeros of $f$ ...
- 33
5
votes
1
answer
453
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Polynomial defined recursively by a resultant
Cross posting from MSE.
Definition:
For any natural number $n\ge 3$, define the polynomial $P_{n}\left(x_1,x_2,...,x_{n-1},x_{n} \right)$, with indeterminates $x_{i}$, where $i\in\{1,2,...,n-1,n\}$, ...
- 149
3
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0
answers
243
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Interlacing sequences by polynomials?
Given $t=2^\ell$ where $\ell\in\mathbb N_{>0}$ and $M\in\mathbb Z$ and two sets of integers $\{a_1,\dots,a_t\}$ and $\{b_1,\dots,b_t\}$ with $0<a_1\leq \dots\leq a_t<M$ and $0<b_1\leq \...
- 13.9k
3
votes
1
answer
138
views
Intersection of quadratic equations with planted solutions?
Suppose we have two quadratic equations in $\mathbb R[x_1,\dots, x_n]$. What is the expected dimension of their intersection?
In general what can we say about intersection of $k$ quadratics? How many ...
- 13.9k
2
votes
0
answers
208
views
Real-rooted polynomials with coefficient constraints
My question is whether there exists $(a_0, a_1, \ldots, a_{2n-1}) \in \mathbb{R}_{+}^{2n}$ such that
(1). $a_{2k} + a_{2k+1} = \binom{3n-1}{3k} + \binom{3n-1}{3k+1} + \binom{3n-1}{3k+2}$ for all $0 \...
- 151
10
votes
2
answers
820
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Simple question about polynomials
Starting from a problem in combinatorics, I ended up with a very simple problem about polynomials, which, unfortunately, I am not able to solve.
Say we work over $\mathbb C$. Fix $d>1$.
Is it ...
- 153
1
vote
0
answers
207
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Polynomial existence over finite field
Denote $\mathcal{F_n}$ as collection of multiaffine polynomials $f\in\Bbb F_2[x_1,\dots,x_n]$.
Denote total degree of $f\in\mathcal{F_n}$ as $deg(f)$ (note $deg(f)\leq n$).
Denote $e_i=(0,\dots,0,\...
- 13.9k
0
votes
2
answers
254
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A specific polynomial triplet question
Notation
$P_k[n]=\{$multilinear polynomials in $\Bbb R[x_1,x_2,\dots,x_{n-1},x_n]$ of total degree exactly $k\}$.
$k=1$ is just linear polynomials.
QUESTION
Is there a triplet $(p,f,g)\in (P_{k}[4]...
- 13.9k
0
votes
0
answers
74
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Upper bound on number of cells created by varieties of co-dimension 1
Say I have polynomials $p_1,p_2,\dots,p_m$ in $\mathbb{R}^n$ (ie. over $n$ variables), each of degree $d$. Is there an upper bound on the number of "regions" created by the surfaces $p_i = 0$? Let's ...
- 143
11
votes
4
answers
1k
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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
".....
- 4,829
8
votes
2
answers
1k
views
What is known about zero-sets of Schur polynomials?
Consider a set S of partitions not containing the empty partition (I would be happy with, say, all the partitions of size less than k, except for the empty one).
Let $U_\lambda^{(r)}$ be the zero-...
- 81