arXiv subjects

Omar Graia

Publications and source records attributed to Omar Graia.

4 recordsLinked to original sources

arXiv2026

The Bi-LF Positive Conjecture Holds

We prove that a semidomain has length-factorial additive and multiplicative monoids if and only if it is isomorphic to the semiring of nonnegative integers or is a unique factorization domain. Consequently, bi-length-factoriality and bi-factoriality coincide for all semidomains, settling the Bi-LF Positive Conjecture. The main step shows that additive length-factoriality and multiplicative factoriality force additive factoriality, without an additive reducedness assumption. Multiplicative factoriality forces every semidomain with a nonzero additive unit to be a unique factorization domain. In the additively reduced case, a uniform multiplier for lattice points in a finitely generated cone yields cancellation of divisibility across sums, which leads to an obstruction to additive atoms dividing positive integers and completes the structural argument. We also consider related questions on quasi-factoriality and classify the rational and transcendental monogenic positive semirings whose multiplicative monoids are quasi-$n$-factorial for a prescribed integer $n \ge 2$.

math.AC↗

arXiv2026

Optimal Deterministic Fully Sparse Matrix Multiplication

We give the first deterministic algorithm for fully sparse matrix multiplication that attains the optimal running-time exponent. This result matches the best previously known randomized algorithm running-time exponent. Given compatible matrices $A$ and $B$ over an arbitrary associative ring with identity, with $\operatorname{nnz}(A),\operatorname{nnz}(B)=O(n^{δ_{\mathrm{in}}})$ and $\operatorname{nnz}(AB)=O(n^{δ_{\mathrm{out}}})$, our algorithm finds the support of $AB$ and computes the product exactly in $$O\!\left(n^{β_R(δ_{\mathrm{in}},\min\{δ_{\mathrm{out}},2δ_{\mathrm{in}}\})+\varepsilon}\right)$$ operations, where $β_R(δ_{\mathrm{in}},δ)$ denotes the maximum of $δ_{\mathrm{in}}$ and $ω_{δ_{\mathrm{in}},R}(a,1,b)$ over all $a,b\in[0,1]$ satisfying $a+b=δ$. For dense inputs over a commutative ring, this bound simplifies to $O(n^{ω_R((δ_{\mathrm{out}}-1)_+,1,1)+\varepsilon})$. With the current rectangular matrix multiplication bounds, this is nearly quadratic, namely $O(n^{2+\varepsilon})$, for every $δ_\mathrm{out}\le1.321334$, improving the previous deterministic range of $δ_{\mathrm{out}}\le 0.642668$. To prove this result, we develop a general deterministic recovery technique that finds and fixes sparse parts of an unknown matrix while keeping temporary errors in denser parts under control.

cs.DS↗

arXiv2026

The Bi-UF Positive Conjecture Holds

A subsemiring $S$ of the nonnegative cone of the real line containing $1$ is called a positive semiring. The Bi-UF Positive Conjecture, posed by Baeth, Chapman, and Gotti in 2021, states that the prototypical semiring $\mathbb{N}_0$ is the only positive semiring whose additive monoid $(S,+)$ and nonzero multiplicative monoid $(S^\bullet,\cdot)$, where $S^\bullet=S\setminus\{0\}$, are both factorial. In this paper, we prove their conjecture.

math.AC↗

arXiv2026

The Bi-UF Positive Conjecture for quadratic monogenic semirings and related progress

A complex semiring is a subset of the complex plane that is closed under the standard addition and multiplication of complex numbers and contains both $0$ and $1$. A complex semiring $S$ is called a bi-UFS if both its additive monoid $(S,+)$ and its multiplicative monoid $(S\setminus \{1\}, \cdot)$ are unique factorization monoids (UFM). The Bi-UF Positive Conjecture states that $\mathbb{N}_0$ is the only subsemiring of the nonnegative cone of the real line that is a bi-UFS. In this paper, we prove that no simple semiring extension of $\mathbb{N}_0$ by a quadratic algebraic number is a bi-UFS, identifying a natural class of complex semirings satisfying the statement of the Bi-UF Positive Conjecture. We also identify another class of complex semirings satisfying the statement of the Bi-UF Positive Conjecture. Then we extend the statement of the Bi-UF Positive Conjecture by motivated by a structural theorem we established for semidomains whose additive monoid are finite-rank free commutative monoids. Finally, we consider the bi-HF property, which is a relaxed version of the bi-UF property. We prove that $\mathbb{N}_0$ is the only positive rational semidomain having the bi-HF property, and we provide two methods to construct bi-HFS complex semirings that are distinct from $\mathbb{N}_0$.

math.GM↗