74 problems

SOS Conjecture. If $A(z,\bar{z})\|z\|^2\in {\rm{SOS}}_n$, then either

Treves' conjecture. The operator $P$ is analytic hypoelliptic if and only if every stratum in the above-described Poisson–Treves stratification is symplectic.

Let $t_y=y(y+1)/2$ and $t_z=z(z+1)/2$ for $y,z\in\mathbb Z$. Sun's conjecture. Any positive integer $n$ is a sum of a square, an odd square and a triangular number; equivalently, e…

Hardy's conjecture. The number of representations satisfies

Let $n$ be an integer, and let $q$ range over the prime divisors of $n$. Girard's conjecture. The integer $n$ is a sum of two squares if and only if every prime divisor $q$ satisfy…

Let $a$ be an integer with $a>30$, and let … Let $S(A(a))$ be the numerical semigroup generated by $A(a)$. For $n\in\mathbb{Z}^{+}$, let $\iota(n)$ be the least number of positive…

Shapiro's field-independence conjecture. The admissibility of a fixed triple $[r,s,n]$ should be independent of the coefficient field, provided the characteristic is different from…

Ternary Pythagoras-number conjecture. More generally, one conjectures that

Superdominance characterization conjecture. The tropicalization $\operatorname{trop}(\mathcal{B}\Sigma_{2d}^*)$ can be described combinatorially in terms of the superdominance orde…

Let $P$ be a nonnegative ternary sextic, let $\Delta_{3,6}$ denote the cone of such forms that are not sums of squares, and let $\delta(P)$ denote its delta invariant. A form is st…

Let $P_{n,d}$ be the cone of nonnegative forms of degree $d$ in $n$ variables, let $\Delta_{n,d}$ be the cone of such forms that are not sums of squares, and let…

Let $P_{3,6}$ be the cone of nonnegative ternary sextic forms, let $\Delta_{3,6}$ denote the cone of such forms that are not sums of squares, and let $\mathcal{E}(P_{3,6})$ denote…

The paper studies number fields $K$ of degree $d$ and the property that $2\mathcal{O}_K^+$ is a sum of squares, where $\mathcal{O}_K^+$ denotes the totally positive elements of the…

Cross-positive decomposition conjecture. Every cross-positive map is a sum of a positive map and a map of the form

Let $\Sigma_{2,8}^4$ denote the cone of sums of squares of binary quartics, and let $f\in\Sigma_{2,8}^4$. A form is doubly positive if it can be written as a sum of two squares who…

Irreducible hypersurface Pythagoras-number conjecture. The following conditions are equivalent:

Let $2d$ be an even degree. A normalized symmetric form that is nonnegative for every number of variables should be a sum of squares. Blekherman’s conjecture. This property holds f…

Let $\mathcal{T}(x)$ count the integers $n\leq x$ such that $b(n)$ is a sum of two squares, where $b(n)$ is the binary partition function. Logarithmic-density conjecture. There exi…

Positive-density conjecture. The set $Q_1$ is infinite. Moreover, $Q_1$ has positive natural density in $\mathbb{N}$. This is suggested by computations showing many such representa…

Let $f\in \operatorname{Sym}^{2d}V$ be generic of SOS-rank $k\leq n$, and let … For a decomposition of $f$ into $k$ squares, write $A\in\mathbb C^k\otimes\mathbb C^N$ for the coeff…

Let $n\ge 3$, let $m_1,\dots,m_n\in\mathbb{N}$, and let $U_1,\dots,U_n$ be independent standard Gaussian random variables. Define the polynomial $H$ on $\mathbb{R}^{n(n-1)/2}$ by ……

Kre1senskmathfrak{d}-Ra61ka-Sgallove1 conjecture. If $\sqrt{2}\in\mathcal{O}$, then

Let $n$ be an integer with $n\geq 2$. The variables $x,y,a,b,c,d$ are non-negative integers, and the two restricted summands are the squares of numbers of the forms $2^a3^b$ and…

Let $K$ be a number field, and let a totally positive element of $K$ mean an element whose image is positive under every real embedding of $K$. Hilbert's four-square conjecture. Ev…

Five-prime Linnik representation conjecture. For every fixed $1<c<c_0$ and every sufficiently large positive number $N$, the displayed Diophantine inequality has a solution in prim…