13 problems

Brown–McMullin conjecture on real reliability roots of simple graphs
arXiv@arxiv
Let $G=(V,E)$ be a connected simple graph, and let $\operatorname{Rel}(G;q)$ be its all-terminal reliability polynomial. Define … For a subset of $\mathbb{C}$ , write an overline fo…
Conjecturerefutedgraph-theory real-roots reliability-polynomials
0
Density conjecture for real reliability roots of graphs
arXiv@arxiv
Let $G$ range over finite connected graphs, and consider the real roots of their reliability polynomials, including the always-present root $q=1$ . The closure of the set of real re…
Conjectureopengraph-theory real-roots reliability-polynomials
1
Non-universality conjecture for the first non-universal term in Kac polynomial real zeros
arXiv@arxiv
Let $P_n(x)=\sum_{k=0}^{n}\xi_kx^k$ be a Kac random polynomial, where the coefficients $\xi_k$ are independent and identically distributed copies of a random variable $\xi$ . Assume…
Conjecturesolvedprobability random-polynomials real-roots
1
Polynomial concentration conjecture for the number of real zeros of Kac polynomials
arXiv@arxiv
Concentration conjecture. Under mild conditions, the probability that the number of real zeros deviates from its expectation by at least $\varepsilon\log n$ is bounded above and be…
Conjectureopenconcentration-inequalities random-polynomials real-roots
1
O. Nguyen's intermediate real-root count conjecture for random polynomials
arXiv@arxiv
O. Nguyen's conjecture. Given $\alpha\in(0,1)$ , there exists a bounded sequence $(c_k)$ of real numbers such that
Conjectureopenprobability random-polynomials real-roots
0
Convergence of the centered variance for derivatives of Kac polynomials
arXiv@arxiv
Variance convergence conjecture. The quantity
Conjectureopenprobability random-polynomials real-roots
2
Distinct positive zeros of the derivative-polynomial combinations
arXiv@arxiv
Polynomial distinct-zeros conjecture. The positive real zeros of the polynomials
Conjectureopenelectrical-fields polynomials real-roots
1
The real $\tau$ -conjecture
arXiv@arxiv
Consider a nonzero polynomial of the form … where each $f_{ij} \in \mathbb{R}[X]$ has at most $t$ monomials. Real $\tau$ -conjecture. The number of real roots of $f$ is bounded by a…
Conjectureopenalgebraic-complexity real-roots sparse-polynomials
0
The conjecture on realizability of pairs with positive positive and negative counts
arXiv@arxiv
Realizability conjecture. The only pairs $(pos,neg)$ that can be non-realizable have either $pos=0$ or $neg=0$ . Equivalently, every pair $(pos,neg)$ satisfying the standard conditi…
Conjectureopenreal-polynomials real-roots sign-patterns
1
Conjectural formula for the expected absolute value of a random polynomial
arXiv@arxiv
Let $\Omega_N$ be the space of monic real polynomials of degree $N$ whose roots lie in the interval $[-1,1]$ , equipped with volume $D_N$ , and let $P$ denote a random polynomial in…
Conjectureopenintegral-formulas random-polynomials real-roots
0
Universality of the limiting constant for expected real roots
arXiv@arxiv
Let $N_n$ denote the number of real roots of a random polynomial of degree $n$ whose coefficients satisfy the assumptions under discussion, and let ${\mathbf{E}}N_n$ be its expecta…
Conjectureopenprobability random-polynomials real-roots
0
Clarkson's conjecture on the number of real roots of the Yablonskii–Vorob'ev polynomials
arXiv@arxiv
Let $Q_n$ denote the $n$ th Yablonskii–Vorob'ev polynomial, defined by $Q_0=1$ , $Q_1=z$ , and … For a real number $x$ , let $[x]$ denote its integer part. Clarkson's conjecture. The n…
Conjectureopenpainleve-equations real-roots special-polynomials
1
Conjecture on $L^1$ convergence of the number of real roots
arXiv@arxiv
Let $G_n$ be the random polynomial considered above, and let $N_{2n}$ and $N_{2n+1}$ denote its numbers of real roots. Suppose that the tail assumptions in the paper hold with…
Conjectureopenprobability random-polynomials real-roots
1