28 problems

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…

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

Linnik's conjecture. For each such $n$, there exists a representation with

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…

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

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…

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 ……

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…

Let $K$ be a totally real biquadratic field, and let $\mathcal{O}_K$ be its ring of integers. The Pythagoras number $\mathsf{P}(\mathcal{O}_K)$ is the least number of squares neede…

Conjecture on ternary approximation. There exists $c_0>1$ such that, for any fixed $1<c<c_0$ and every sufficiently large positive number $N$, the inequality

The computational refinement of the 1-3-5 conjecture. Except for $m\in\{31,43,111,151,168,200,248,263,319,456,479,871,1752,1864,3544\}$, every such $m$ has a representation for whi…

Sun's refined 1-3-5 conjecture. Any positive integer can be represented in this way with $x+3y+5z$ a square, and with at least one of the following properties: $x$ is three times a…

Let $\varepsilon>0$. For each sufficiently large odd squarefree integer $n\not\equiv7\pmod{8}$, consider representations of $n$ by three integer squares. Linnik's conjecture. There…

Let $\mathbf t=(t_0,\ldots,t_{d-2})$ and … where $N,b_0,\ldots,b_{d-2}$ are integers, $N\equiv\sum_{i=0}^{d-2}b_i^2\pmod q$, and $\gcd(N,q)=1$. For $r>0$, let … The sum-of-two-squa…

Let $P_{n,4}$ denote the space of homogeneous quartic forms on $\mathbb{R}^n$, and let $G\in P_{n,4}$ be positive semidefinite if $G(x)\geq 0$ for all $x\in\mathbb{R}^n$. Suppose t…

For even $n$ and even degree $m$, let $P_{n,m}$ be the cone of non-negative real homogeneous forms in $n$ variables of degree $m$, and let $\Sigma_{n,m}$ be the subset consisting o…

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

Let $n\ge3$, $d\ge2$, and define … Let $Z$ be a set of $s$ general points in $\mathbb{P}^{n-1}$, where $N_{d-1}\le s<N_d$, and let $I=I(Z)$ be its vanishing ideal. Write…

Let $P(z)=(P^1(z),\ldots,P^q(z))$ be a polynomial mapping in $z=(z^1,\ldots,z^n)\in\mathbb C^n$, and let $A(z,\bar z)$ be a Hermitian polynomial satisfying … If $r$ denotes the lin…

Ottaviani–Shapiro conjecture. For any number of variables,

Linnik's microsquares conjecture. Whenever $n$ is sufficiently large in terms of $\varepsilon$, this equation possesses a solution in which

Let $h$ be a hyperbolic polynomial in direction $e$, let $\mathcal A_e(h)$ be its generalized Clifford algebra, and let $\Sigma^2\mathcal A_e(h)$ denote the cone of sums of hermiti…

Real-point conjecture. For $p \in P_{3,6} \setminus \Sigma_{3,6}$ there exist $v_1,\ldots,v_9 \in \mathbb{R}^3$ yielding a separating extreme ray for $p$. Analogously, for…

Invariant sums-of-squares induction conjecture. The induced quadratic module satisfies