We study the exact learnability of finite unions of intersecting affine modules in one dimension. An affine module is a set of the form a+∑_j=1^sb_j ℤ, where a,b_1,…,b_s∈ℕ. We say that a set definable as a finite union of affine modules is a union of intersecting affine modules if it admits a representation in which all modules have a non-empty intersection. We show that this class is efficiently exactly learnable using equivalence and subset queries. Moreover, subset queries can be replaced with membership queries when a common element is known. Our algorithm requires at most klog(2|x_ℓ|)+2k counterexamples, where k is the number of affine modules in the smallest representation and x_ℓ is the largest counterexample. This implies polynomial-time learnability in the binary representation.
We study the problem of computing the isolated regular solutions of a system (f_1,…,f_n) of n polynomial equations in n variables (X_1, …, X_n) over a field of characteristic zero k. We focus on systems with a composable structure, where each polynomial f_i can be expressed as a composition f_i = h_i(g_1,…,g_n). Exploiting this structure allows us to reduce the original system to one in the g_j variables, thereby significantly improving the efficiency of symbolic solution algorithms. We present a probabilistic algorithm that computes all isolated regular solutions, with arithmetic complexity being polynomial in the input size and in the number of solutions. A first important application is when f_1, …, f_n belong to the subring k[g_1, …, g_n], where g_1, …, g_n are algebraically independent polynomials in k[X_1, …, X_n]. Another important application is to systems of invariant polynomials under finite reflection groups, since by the Chevalley-Shephard-Todd theorem their invariant rings are polynomial algebras. Typical examples include the symmetric groups S_n, the hyperoctahedral groups B_n, the dihedral groups I_2(m), and the exceptional finite reflection groups E_6, E_7, E_8, F_4, H_3, H_4.
When a computer algebra system fails to solve an Ordinary Differential Equation, is this a limitation of its implementation, or a genuine computational barrier? Three traditions bear on the question. Modern computer algebra algorithms can be extremely efficient: Newton-type methods solve polynomial ODEs over ℚ[[X]] in quasi-linear time. Analog models of computation has shown that polynomial ODEs and Turing machines are two presentations of the same phenomenon, with solution length acting as time and precision as space. Computable analysis shows that ODEs can be intrinsically hard – undecidable, even 𝖯𝖲𝖯𝖠𝖢𝖤-complete, over compact domains. Comparing these traditions is natural and necessary, yet such comparisons routinely reduce to comparisons of encodings rather than of underlying algorithmic content. We argue that reverse mathematics provides a representation-invariant lens in which algorithmic content is compared directly. We prove that every level of the Big Five hierarchy is inhabited by a natural statement from classical ODE theory, as an exact equivalence: the regularity of f is an intrinsic algorithmic invariant placing the initial value problem y'(t)=f(t,y(t)), y(t_0)=y_0, into one of several computational strata, ranging from polynomial-time solvability to transfinite computation. The resulting stratification acts as a practical diagnostic common to the three traditions. By abstracting from representation, it separates fundamental barriers from the technical shortcomings of symbolic solvers, the artefacts of analog encodings, and the effectivity constraints of computable analysis, identifying the intrinsic parameters (length bounds, radii of convergence, moduli of continuity) under which feasibility is restored.
Cylindrical Algebraic Decomposition (CAD) algorithms typically produce a decomposition adapted to a finite family of semi-algebraic sets ℱ (i.e. every member of ℱ is a union of cells). Different algorithms may produce different outputs, and introduce unnecessary cell divisions. Recent work by Michel, Mathonet, and Zénaïdi in ISSAC 2024 formalised this issue by studying the refinement order on the set of all CADs adapted to ℱ and analysing the existence of a minimum (coarsest) adapted CAD. It was shown that such a minimum adapted CAD always exists for subsets of ℝ and ℝ^2, but not of ℝ^n (n ⩾ 3) in general. It is natural to seek natural classes of subsets of ℝ^n that admit a minimum adapted CAD. In this paper, we identify a class of subsets of ℝ^3 that contains all algebraic sets for which minimum adapted CADs do exist. This provides the first positive existence theorem for minimum CAD for a non-trivial class of sets.
This paper focuses on asymptotic properties of random monomial ideals through a statistical viewpoint. It extends the study of redundancy in monomial ideals by analyzing the poset density of the LCM-lattice. We explore how this density behaves across random algebraic models and structured networks. Experimental data reveal that the LCM-lattice exhibits sharp threshold behavior rather than changing smoothly. We observe a strong negative correlation between the number of generators and LCM-lattice density, abruptly separating three distinct regimes: a low-density Taylor-like regime, a high-density redundant regime, and a narrow transition window. We show that increasing the generator degree causes this density drop to occur at lower probability thresholds. We conclude by conjecturing that for equigenerated squarefree ideals, the LCM-lattice density undergoes a sharp phase transition, analogous to the emergence of giant components in hypergraphs. This suggests that the classical, ideal-by-ideal role of the LCM-lattice as a combinatorial invariant also admits a statistical/asymptotic counterpart: in natural random families, redundancy and resolution-complexity indicators concentrate into distinct typical regimes separated by a narrow transition window.
We analyze the sum-of-squares rank of unweighted instances of the Minimum Knapsack (MK) problem, i.e., minimization of ∑_i=1^n x_i for 0/1 variables under the constraint ∑_i=1^n x_i ≥ q, with q ∈ℝ. Such instances have long served as a testbed for understanding the limitations of lift-and-project methods in Boolean optimization. For example, both the Lovász-Schrijver and Sherali-Adams hierarchies require (maximal) rank n to solve them, already when q=1/2 is constant. The SOS hierarchy requires only sublinear rank O(√(n)) to solve unweighted MK when q=1/2. On the other hand, when q is allowed to vary with n, the SOS rank of the problem may become linear. Interestingly, this is known to happen both when q is large, and when q is very small (0<q ≤ 2^-n). This raises the question of whether we should think of hard instances of unweighted MK as being typical for the SOS hierarchy, or as a consequence of very specific choices of the threshold parameter q. In this paper, we address this question by showing new upper and lower bounds on the SOS rank of unweighted MK in the whole regime of the parameter q. For n-q ≤ O(1), we show that the SOS rank is constant. In contrast, when q ≤ O(1), a linear rank is needed if q is exponentially close to an integer. As our main positive result, we show that linear rank is very rare for q ≤ O(1). This can be expressed in the language of smoothed analysis: after perturbing q by a Gaussian with mean 0 and variance σ^2, the expected SOS rank of MK is O(√(n)log (n/σ)).
Connected components of real algebraic sets are semi-algebraic, i.e. they are described by a boolean formula whose atoms are polynomial constraints with real coefficients. Computing such descriptions finds topical applications in optical system design and robotics. In this paper, we design a new algorithm for computing such semi-algebraic descriptions for real algebraic curves. Notably, its complexity is less than the best known one for computing a graph which is isotopic to the real space curve under study.
