34 problems

Let a $GC_n$ set be an $n$-correct set $\mathcal{X}$ in which the fundamental polynomial of every node is a product of $n$ linear factors. Gasca–Maeztu conjecture. Any $GC_n$ set c…

Let $P$ be a finite-rank linear projector on $4\mathbb{C}[\bm{x}]4$. An ideal projector is a linear idempotent operator whose kernel is an ideal; a Lagrange projector is an ideal p…

Let $(A,S)$ be a Birkhoff interpolation scheme, with $S$ a lower set, and fix non-negative integers $p$ and $q$. An inverse shift moves elements of $A$ downwards and to the left wi…

Let $(A,S)$ be a Birkhoff interpolation scheme, where $A$ is a finite set of derivative indices and $S$ is a lower set. Fix non-negative integers $p$ and $q$, and let $(p,q)$-recta…

A lower set $S$ is a finite subset of the non-negative integer lattice such that, whenever $(\beta_1,\beta_2)\in S$ and $0\leq\alpha_i\leq\beta_i$ for $i=1,2$, one has…

Let $n,m\in\mathbb{N}$, with $m=\lfloor\theta n\rfloor$ for a fixed $\theta\in(0,1)$. Let $\Lambda_n^m$ and $\tilde\Lambda_n^m$ be the Lebesgue constants associated with the interp…

Self-conjugate-grid uniform conjecture. For every $0\leqslant k\leqslant n-1$,

Schur-polynomial pointwise conjecture. For all $0\leq k<n\leq t$,

Self-conjugate-grid pointwise conjecture. For every $0\leqslant k\leqslant n-1$,

Kallioniemi's conjecture. In each complementary segment $J_s$, the function $\operatorname{\mathsf{sup}}_{g\in\mathscr{L}_n(I)}|g^{(k)}(\cdot)-\hat g^{(k)}(\cdot\mid z_{1:n})|$ is…

Dubiner-metric equidistribution conjecture. Equidistributed nodes with respect to this distance are near-optimal for bivariate interpolation. This conjecture concerns the choice of…

Strong Bergman asymptotics. The measures satisfy

Let $F(x,y)$ be the integer-coefficient polynomial corresponding to the numerically computed polynomial curve that passes through the 16 line segments, with its coefficients normal…

Hesthaven's conjecture.

Consider the polynomial-interpolation approach to computing the volume of a restricted independent set polytope. The approach encounters a pitfall even for a one-vertex graph, beca…

Let the Morrow–Patterson points be the interpolation points on the square supporting a minimal positive cubature formula with $N$ nodes of degree of exactness $2n$ for the product…

Let $S_4$ be the real four-dimensional simplex, and let $F_5$ be the set of $126=\binom{9}{4}$ points constructed by placing the specified boundary points on every face and the rem…

Let $S_3$ be the real three-dimensional simplex, and let $F_5$ be the set obtained by taking the boundary points described in the paper together with the four interior points … max…

Let $S_d$ be the real $d$-simplex, let $F_4$ be the set of $N=\binom{d+4}{4}$ points constructed in the paper, and let $\ell_j$ be the associated fundamental Lagrange polynomials.…

Let $S_d$ be the real $d$-simplex, let $F_4$ be the set of $N_4=\binom{d+4}{4}$ points constructed in the paper, and let $\ell_i$ be its fundamental Lagrange polynomials. The degre…

Let $H_{6,m}$ have Fourier expansion … where $q_m(z)=\exp(2\pi i z/\lambda_m)$ and $m\geq3$. Brent's $H_{6,m}$ interpolation conjecture. For every $n$, there is a polynomial…

Let $H_{4,m}$ have Fourier expansion … where $q_m(z)=\exp(2\pi i z/\lambda_m)$ and $m\geq3$. Brent's $H_{4,m}$ interpolation conjecture. For every $n$, there is a polynomial…

Let $m\geq 3$, let $G(\lambda_m)$ be the Hecke group with $\lambda_m=2\cos(\pi/m)$, and write the Fourier expansion of the normalized invariant as … where…

Assume that the Gasca–Maeztu conjecture holds for all degrees up to $n$. Let $\mathcal X$ be a $GC_n$ set, let $\ell$ be a $k$-node line with $2\le k\le n+1$, and let…

Singularity conjecture. For every $k\in\mathbb{N}$ and every such matrix $(\alpha_{ij})$, the associated sample matrix is singular.