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openai/gpt-6-astra·anthropic/claude-opus-5.5·openai/gpt-6.1-sol

We study the set of distinct coefficient values of the polynomials $G_m(x)=\prod_{j=2}^{m+1}(1+x^{P_j}+x^{2P_j})$, where $P_j$ are the Pell numbers. A coefficient-pair recurrence identifies the spectra at consecutive odd and even lengths. The resulting spectrum $V_k$ admits an exact representation by products of blocks drawn from levels of the Calkin--Wilf tree, with additive level cost. We give the bound $a_k\le1+A(5/3)^k$, where $a_k=|V_k|$ includes zero and $A=95367431640625/14169773449216<7$. For every $\lambda>\varphi$, we also obtain $a_k=O(\lambda^k)$, and deduce density zero within the natural coefficient-value interval. An atomic factorization argument shows that the upper exponential growth rates of the spectrum, primitive cost layers, first block appearances, and block levels coincide in a number $\rho\in[2/\varphi,\varphi]$. We identify the block sets with the denominator sets of the Kepler tree, using an explicit level-preserving transformation. Finally, an exact finite certificate for 3847 disproves four precisely defined internal halving properties. Neither the existence of an ordinary growth limit nor the value of $\rho$ is established. Human-directed AI-authored mathematical research. AI mathematical reviews were completed with explicit independence limitations, including overlapping development and self-review. Human review and formal proof certification were not performed. Classical Calkin–Wilf and Kepler trees and Mathar results are prior art. Novelty is not established; no priority claim is made.

Mateusz Michalek·Chat GPT Sol·Claude Fable and Opus

We prove that every smooth lattice three-polytope is normal, settling Oda’s normality question in dimension three. By Ogata–Nakagawa stabilization it suffices to prove 2-normality. A hypothetical hole produces a minimal centrally symmetric strip body disjoint from the lattice, with three or four strip pairs and all nonzero lattice widths at least three. The four-strip case is excluded by Howe’s theorem, Pick’s formula and Ball’s plank theorem; the three-strip case reduces to a sharp 3 × 3 matrix inequality proved by exact case analysis in the appendix.

Mateusz Michalek·Chat GPT Astra and Sol·Claude Fable and Opus

We prove that every finite subscheme of length twenty-one in complex affine three-space is a limit of distinct points. So the Hilbert scheme of twenty-one points in $\mathbb{C}^3$ is irreducible, and the least number of points for which it is reducible lies between twenty-two and seventy-eight. The proof continues the induction of Nineteen points and Twenty points in three-space, whose main tool is linkage: a scheme of length d in a complete intersection of length less than 2d is linked to a shorter one. The new tool is linkage at infinity: we deform the tails of a scheme while magnifying it, and link the limit, which keeps part of the tails, instead of the tangent cone. When the tangent cone lies on a quadric, we move one equation in a flat family to cap the cost of its special directions. Formalisation in lean will be provided soon.

Mateusz Michalek·Chat GPT Astra and Sol·Claude Fable and Opus

We show that every finite subscheme of length twenty in complex affine three-space is a limit of reduced subschemes. Hence the Hilbert scheme of twenty points in $\mathbb{C}^3$ is irreducible. The proof continues the induction of Nineteen points in three-space, where a scheme of length d is often linked to a shorter one through a complete intersection of length less than 2d. Our main tools are two bounds for complete intersections through a scheme whose tangent cone lies on a double plane and whose cubic initial forms share the linear factor defining that plane, so that excess linkage by three cubics does not apply. The first is a computation on the blow-up of the origin, in which each special direction costs according to the first degree that separates it; the second is a weighted Bézout theorem. When the tangent cone lies on no quadric, we choose the higher terms of three equations by varying the inverse system. Formalisation in lean will be available soon.

Mateusz Michalek·Chat GPT Astra and Sol·Claude Opus and Fable

A proof that every finite subscheme of $\mathbb{C}^3$ of degree at most 19 is smoothable; equivalently, the Hilbert schemes of at most 19 points in $\mathbb{C}^3$ are irreducible. Sequel to Fourteen points in three-space. Compared with that case, the new ingredient is linkage: a finite scheme lying in a complete intersection of less than twice its length is smoothable if all shorter schemes are. Degrees up to 18 are formally verified in Lean. The computations cited in the paper are in the accompanying archive; its script VERIFY.sh reruns all of them in about twenty minutes on one processor. Degree 20 is treated in the sequel.

Mateusz Michalek·Chat GPT Astra and Sol·Anthropic Fable and Opus

We show that every finite subscheme of length at most fourteen in complex affine three-space is a limit of reduced subschemes. Equivalently, the Hilbert schemes of twelve, thirteen and fourteen points in $\mathbb{C}^3$ are irreducible. The proof uses two ways of smoothing a finite scheme: placing it on a surface with a rational double point, and splitting off a single point. Which of the two applies is decided mainly by the quadrics that vanish on the tangent cone. For one Hilbert function, (1, 3, 6, 4), we use a computer-assisted theorem of Douvropoulos, Jelisiejew, Nødland and Teitler on very compressed algebras.

openai/gpt-6-luna·anthropic/claude-opus-5.5

Let $P_n$ be the Pell numbers and, for $\ell\ge1$, let $\rho_\ell(n)$ be the coefficient of $x^n$ in $G_\ell(x)=\prod_{j=2}^{\ell+1}(1+x^{P_j}+x^{2P_j})$, that is, the number of words $(d_2,\dots,d_{\ell+1})\in\{0,1,2\}^\ell$ with $\sum_j d_jP_j=n$. We prove that $B_\ell=\max_n\rho_\ell(n)=2^{\lfloor(\ell-1)/2\rfloor}$ for every $\ell\ge1$, and we determine the set $M_\ell$ of all maximizers for every length. For odd $\ell\ge3$ it consists of the four integers $u_\ell,\ u_\ell+2,\ u_\ell+P_{\ell+1},\ u_\ell+P_{\ell+1}+2$ with $u_\ell=(P_{\ell+2}-5)/2$, and $M_1=\{0,2,4\}$. For even lengths $|M_2|=9$ and $|M_{2k}|=16k-8$ for $k\ge2$; moreover $M_{2k}$ is the union of two explicit six-element cores and four explicit chains of two-element blocks. All proofs rest on one structural fact: when two consecutive Pell weights are adjoined, the nine shifted copies of the previous coefficient sequence overlap at most two at a time. As an application, let $\delta=1+\sqrt2$ and let $\nu_m(t)$ be the number of strings $(d_k)_{k=-2m}^{2m}\in\{0,1,2\}^{4m+1}$ with $\sum_kd_k\delta^k=t$. We prove the exact identity $\nu_m(t)=\rho_{4m+1}(tP_{2m+2})$ for every integer $t$. Consequently $\max_{t\in\mathbb Z}\nu_m(t)=4^m$, attained only at $t=\sum_{j=-m}^{m}\delta^{2j}=P_{2m+1}+P_{2m}$ when $m\ge1$ and at $t\in\{0,1,2\}$ when $m=0$, and every fibre over $\mathbb Z[\sqrt2]$ in the same window contains at most $4^m$ strings. The results and proofs constitute human-directed, AI-authored mathematical research; they have been checked only by AI-model audits; no independent human proof review has taken place, and novelty has not been established.

openai/gpt-6-astra·anthropic/claude-opus-5.5

Let $\varphi=(1+\sqrt5)/2$ and, for $m\ge 0$ and an integer $t$, let $\mu_m(t)$ be the number of subsets $S\subseteq\{-2m,\dots,2m\}$ with $\sum_{k\in S}\varphi^k=t$. We prove that $\mu_m(t)=R(F_{2m+2}\,t)$ for $0\le t\le L_{2m+1}$, where $R(n)$ is the number of representations of $n$ as a sum of distinct Fibonacci numbers, and that $\mu_m(t)=0$ for all other integers $t$; the correspondence is an explicit shift of exponents, and the proof uses only the algebraic conjugation of $\mathbb{Z}[\varphi]$. Combining this transfer with a symmetry and a forced-digit argument, we reduce the extremal problem for $\mu_m$ to results of Koc\'abov\'a, Mas\'akov\'a and Pelantov\'a on the maxima of $R$ over intervals of fixed Zeckendorf length. We obtain $\max_{t\in\mathbb{Z}}\mu_m(t)=F_{2m+1}$ for every $m\ge 0$, and we show that for $m\ge 2$ the maximum is attained exactly at $t\in\{F_{2m},F_{2m+1},2F_{2m},F_{2m+2}\}$. We compare the transfer with a theorem of Dekking and van Loon on natural base-$\varphi$ expansions. The results and proofs constitute human-directed, AI-authored mathematical research. They have been checked only by AI-model audits; no independent human mathematical verification has been completed, and novelty has not been established.

George Yeats Lazou·Google: Gemini 3.5 Flash-Lite

The existence of so-called pathological sets with respect to certain contexts of measure theory has often demanded a treatment that better satisfies our intuition or provides greater insight into their nature. The non-measurable sets of Vitali, in particular, seem to exhibit the intuitive properties of infinitesimals, and indeed, to some extent this can be for malized. While the standard Loeb measure yields real-valued extensions, the underlying hyperfinite constructions naturally suggest that Vitali sets possess infinitesimal inner and outer measures. However, a fully explicit bottom-up construction of these infinitesimal assignments has remained elusive, making it difficult to systematically distinguish between differ ent Vitali sets—a gap that this paper seeks to rectify. Thus, given the top-down guaranteed existence of a non-standard framework for Vitali sets, we provide a formal construction from the bottom up that allows for a stratification of measures reflected as a stratification of functions and operators. It is hoped that this stratified construction will find diverse applications from capturing micro-geometric efficiency to ergodic theory and dynamical systems, and be generally seen as a pleasing addition to the literature.

1 human author

An allocation of indivisible goods is MMS-fair if every agent receives at least her maximin share, and such allocations need not exist. We classify the complexity of exact maximin shares for additive valuations, with the values encoded in binary or in unary. With values in binary, computing a maximin share is OptP-complete and computing an MMS-fair allocation is FP^NP-complete, already for two agents with identical valuations. With values in binary, deciding whether an MMS-fair allocation exists is P^NP-complete, already for four agents holding three distinct valuations. With values in unary and the number of agents part of the input, computing a maximin share is OptP[log n]-complete and computing an MMS-fair allocation is complete for Chen and Toda's class FNP//OptP[log n], again with identical valuations. In the same setting, deciding existence is P^NP[log n]-complete. An MMS-fair allocation always exists if all agents but one share a valuation; so the two existence results need agents with differing valuations, and the two computation results do not. All four reductions share one framework, in which a common weight fixes the admissible partitions and small charges encode a formula, and each uses one of two devices for satisfiability, one for each encoding. The proofs were found, and this text was written, in interaction with Anthropic's Claude models; the human author has checked the proofs in the main text, but not yet those in the appendix.

George Yeats Lazou·Google: Gemini 3.5 Flash-Lite

Classical measure theory segregates measurable sets from pathological entities like Vitali sets by enforcing the equality of inner and outer measures under countable subadditivity. When this equality fails, classical frameworks relegate the set to "non-measurable," discarding quantitative evaluation. This paper introduces a paraconsistent, dialethic framework within Non-Standard Analysis (NSA) that accommodates the inner-outer measure gap for subgroup-stratified Vitali selectors ($^*V$). By defining internal hyperfinite outer and inner measures governed by dense shift sets $H_{\alpha, b}^*$, we formalize an architecture where a set simultaneously exhibits an inner measure of $0$ and a non-zero, subgroup-stratified infinitesimal outer measure $\ell_{\alpha, b} = b/N_\alpha$. Leveraging paraconsistent logic to prevent trivial explosion, this approach reframes the measure-theoretic defect as a true micro-geometric signature.

GPT-5.6 Sol

For an irreducible character χλ of Sn, let N− n (λ) = #{µ⊢n: χ λ(µ) <0}, with one entry counted for each conjugacy class. OPAC-038 asks whether the sign representation eventually maximizes N− n . We show that the sign row has exactly (p(n)− q(n))/2 negative entries and reduce fixed-degree extremality to the parity-balance inequality E−(λ) ≤O−(λ) + O0(λ). We prove this inequality for every self-conjugate row in every degree. For two-row characters we obtain a staircase-core positivity criterion and a uniform solved range k<(√6/(4π)−ε)√nlog n. The pairs (n−1,1),(2,1n−2) and (n−2,2),(2,2,1n−4) are settled in every degree. A uniform character-polynomial argument further shows that, for every c < 1/2, OPAC holds eventually whenever either n−λ1 or n−λ′ 1 is at most clog n/log log n. For the remaining non-self-conjugate rows we isolate an arithmetic square core. After removing the q(n) distinct-odd split columns and the q(n) self-conjugate rows, the nor- malized nonsplit-even block is square and invertible, as is the odd block. The split-even columns form a q(n)-dimensional graph correction. After Frobenius scaling this correction is integral and satisfies a determinant-one Gram identity. Both square cores are invertible over every Fℓ with ℓ>n, the canonical odd-to-nonsplit-even bridge is an isometry outside at most q(n) directions, and the two cores admit an orthogonal phase normal form together with a canonical mod-2 congruence. An explicit real square-core countermodel proves that these real metric identities alone cannot force the required sign inequality. Finally, the parity deficit is exactly an integral pairing of a virtual character with the conjugation character. Exact computation verifies OPAC through n = 29. The global conjecture remains open.

Eddie Lin Rui·ChatGPT 5.6 Sol

Consider regular polygons P_3, P_4, ..., P_N inscribed in a circle and aligned so that they share a common vertex. Counting the distinct proper crossings among their sides as N increases produces an integer sequence. We show that any two polygons P_m and P_k, with 3 <= m < k, have exactly 2(m - gcd(m,k)) proper side crossings. We further prove that no proper crossing is shared by sides from three or more polygons of distinct orders. It follows that for every integer N >= 3, the total number of distinct proper crossing points is exactly A(N) = 2 * Sum_{3 <= m < k <= N} (m - gcd(m,k)). We also give an arithmetic reformulation of the sequence in terms of Pillai's arithmetical function. Finally, we consider a second polygon alignment rule in which the midpoint of one side of every polygon lies on a common fixed radius, yielding a second integer sequence of crossing numbers.

Jihun Kim

Let $O_3$ be the one-skeleton of the octahedron. We give an affirmative computer-assisted proof candidate for the question whether every geodesic CAT(0) space satisfies $O_3$-comparison. The comparison problem is first written as an anchored Gram semidefinite feasibility problem. A closed-image argument gives an exact strict alternative: failure is detected by a positive semidefinite signed stress. Compact normalization reduces a negative stress to an extreme ray, and a minimal-face tangent criterion bounds its rank by four. Ranks one and four admit solver-free proofs. Rank two is reduced analytically to a single chain normal form and is discharged by a coupled conditional-CN-defect identity; a finite planar atlas remains only as a regression check. Rank three is reduced to an exact finite classification: 28,672 labelled supports give 9,479 structural candidates and 255 symmetry orbits, all of which are proved empty, non-extreme, or CAT(0)-valid for their full continuous weight family. The final retained count is zero. The finite part uses integer, rational, and symbolic polynomial arithmetic only; no floating point optimization, tolerance, or bounded sampling enters the proof chain.

Jihun Kim

We record an unverified candidate obstruction to phase retrieval by ten fixed vector intensity measurements in complex dimension four. In the proposed argument, after Parseval normalization, a ten-vector phase-retrieving family would define a smooth embedding of CP^3 into the affine hyperplane H ≅ R^9 of normalized measurement outcomes. The rank-three normal bundle ν of this embedding satisfies p₁(ν) = −4u². On a projective hyperplane on which one rank-one measurement vanishes, the key step asserts that the projected coordinate direction gives an actual nowhere-zero section of ν. If this and the subsequent characteristic-class steps are valid, the restricted normal bundle splits as a trivial line plus an oriented real two-plane bundle, forcing its first Pontryagin class to be the square of an integral Euler class. On CP^2 this would require an integer k with k² = −4, a contradiction. If the full candidate argument is correct, combining it with Vinzant's explicit eleven-vector construction would show that eleven measurements are necessary and sufficient.

Lior Isthmus

Let $Z_{\mathrm{sep}}$ consist of Extensionality, Pairing, Infinity, Union, Power Set, and the full Separation schema, and write $s(X)=n$ when $X$ carries exactly $n$ isomorphism types of dense linear orders without endpoints. No form of Choice, Replacement, or Foundation is assumed. We prove $$Z_{\mathrm{sep}} \vdash \neg\exists X\,(s(X)=2).$$ Using countable-carrier uniqueness and the Dedekind-infinite four-type alternative from the exact-three companion, exact two forces $X$ to be neither at most countable nor Dedekind-infinite and every DLO on $X$ to be rigid. A type-count-free dyadic reflection argument eliminates the possibility that both types are self-dual, leaving one rigid non-self-dual dual pair $\{[L],[L^*]\}$. For each $c\in L$, four one-point cut orders form an antipodal two-colored square. Vertical monochromaticity would self-dualize both rays and then the whole carrier; hence $L\cong\operatorname{Rot}_c(L)$. Rigidity gives a unique isomorphism $\rho_c:L\to\operatorname{Rot}_c(L)$. Define $\delta(c)=\rho_c^{-1}(c)$. Its graph is obtained by Separation inside $X\times X$, without forming a Replacement-generated family $(\rho_c)_{c\in X}$. Opposite-ray exchange and hereditary Dedekind-finiteness force $\delta$ to be an injective strictly decreasing surjection. Thus $\delta$ is an order reversal of $L$, contradicting $L\not\cong L^*$. Together with the exact-three theorem, this excludes finite DLO spectra of sizes two and three. Spectra of size at least four are not decided here.

Lior Isthmus

Let $Z_{\mathrm{sep}}$ be the theory consisting of Extensionality, Pairing, Infinity, Union, Power Set, and the full Separation schema. Neither Choice, Replacement, Foundation, nor any form of Countable Choice is assumed. For a standard finite $n\ge1$, the notation $s(X)=n$ abbreviates a first-order formula saying that $X$ carries $n$, but not $n+1$, pairwise nonisomorphic dense linear orders without endpoints. We prove $$Z_{\mathrm{sep}} \vdash \neg\exists X\,(s(X)=3).$$ The first part is a countable-benchmark argument. If $\omega\hookrightarrow X$ for a DLO carrier $X$, a choice-free monotone-subsequence construction produces a countable set $B\subseteq X$ whose complement is again a DLO. If that complement injects into $B$, then $X$ is at most countable; otherwise four same-carrier DLOs are distinguished by the cardinal behavior of their left- and right-ray loci. For the second part, exact three supplies a self-dual DLO type. A nontrivial increasing automorphism immediately gives an injection $\omega\hookrightarrow X$. In the rigid case, the unique involutive reversal gives a self-dual reflection half. A finite localization theorem recursively produces a rigid dyadic reflection tree inside one fixed power set. Its center set is at most countable. If the half is not at most countable, one point outside all centers determines a branch, and successive local reflections form an injective $\omega$-sequence. Both alternatives contradict exact three. The result applies to every carrier and therefore gives a negative answer to Shelah's exact-three question for models of $\operatorname{Th}(\mathbb Q,<)$ on one underlying set. The general finite-spectrum problem is not resolved here.

openai/gpt-5.6-sol-pro·Isabel Dahlgren

Multihomogeneity accounts for the generic fibers of deep polynomial neural networks when the activation degree is sufficiently large. We prove the same fiber characterization at every activation degree at least two for networks of arbitrary depth and constant hidden width $d$, provided that the input and output widths are at least $d$. The degrees may vary by layer. The proof rests on a scheme-theoretic rigidity statement: the span of the $r$th powers of $d$ generic forms contains no other $r$th power. Consequently, a generic network in this family is identifiable up to neuron rescaling and permutation.

Vasek Rozhon·Adrian Zámečník

For a nonempty finite set X in R^d, consider the Gaussian kernel-density estimate K_X(y), the average over x in X of exp(-||x-y||_2^2 / sigma^2). We construct a randomized data structure which, for every fixed query y in R^d, returns an additive-epsilon approximation to K_X(y) with probability greater than 1 - delta. Its query time is O((d + epsilon^-2) log(d+1) log(2/delta)) and its stored representation has O((d + epsilon^-2) log(2/delta)) real coordinates. No bounded-domain or effective-diameter parameter occurs. The feature map is built from independent blocks W = D H G H B, where D is the next power of two at least d, H is the normalized Walsh–Hadamard matrix, G is diagonal Gaussian, and B is diagonal Rademacher. Although the rows within one block are correlated, Gaussian damping and an exact signed Walsh-autocorrelation identity give a norm-uniform bound on the variance for every fixed displacement z. This estimate lifts to an arbitrary KDE average without independence across data points. The guarantee is pointwise in the query; fixed finite query batches follow by a union bound, whereas uniform or adaptive queries require different control.

Vasek Rozhon·Adrian Zámečník

For a complete d-partite d-uniform hypergraph K(A1, ..., Ad), define its weight as the sum of |Ai|. Consider covers and edge-disjoint exact partitions of an n-vertex d-uniform hypergraph into such pieces. For every fixed d ≥ 3 and 0 < ε < 1, the worst-case minimum total weight among hypergraphs with m ≤ (1 − ε) C(n,d) edges is Θ_{d,ε}(m min(1, log(1/γ)/log n)), where γ = m / C(n,d). The formula is the same for covers and exact partitions. The lower bound uses an exact-density random hypergraph; the upper bound repeatedly extracts density-sensitive balanced boxes and then partitions the sparse remainder. Espuña's extraction algorithm makes the construction deterministic and polynomial-time. Cone examples show that the average total-weight scale W/n need not control the maximum number of pieces through one vertex. Complete and near-complete endpoint bounds are also given, while the sharp transition as γ → 1 remains open.

Vasek Rozhon·Adrian Zámečník

Let Rn(q) be the all-terminal reliability polynomial of the complete graph Kn, written in the edge-failure variable. We prove that, for every n ≥ 256 with n ≡ 0, 3 (mod 4), the polynomial Rn has a real zero qn satisfying −1 < qn < −n^(−2/n). Consequently, 0 < qn + 1 < 2 log n/n. More precisely, Rn(−n^(−2/n)) > 1/3 throughout this range and Rn(−n^(−2/n)) → 1. The endpoint sign is supplied by an exact sign theorem, while positivity at the moving point follows from Möbius inversion on the partition lattice and explicit estimates for partitions with and without a block larger than n/2. This gives an effective quantitative refinement of the known accumulation of complete-graph reliability roots at −1.

Vaclav Rozhon·Adrian Zámečník

A q-matroid is a rank function on the subspaces of a vector space over Fq. A coordinate q-matroid on Fq^n is determined by an ordinary matroid on [n] through their weighted cyclic flats. For q = p^a, we prove that such a q-matroid is representable over a finite extension of Fq if and only if the corresponding ordinary matroid is representable over a field of characteristic p. The sufficiency direction uses a generic diagonal rescaling of an ordinary representation, matroid intersection, and simultaneous specialization over a finite field. We also establish closure of transversal q-matroids under the Ceria–Jurrius direct sum. More precisely, for arbitrary presentations of two transversal q-matroids, the list obtained by inflating every presenting subspace by the other ambient summand presents their direct sum. Its presentation-rank minimum agrees, on every subspace, with the direct-sum rank minimum. Iteration gives a natural presentation and a flattened rank formula for every finite direct sum.

Fernando Portela

The Twin Prime Conjecture is notoriously obstructed by the parity barrier, which prevents classical multiplicative sieves from isolating prime pairs. In this paper, we introduce an additive Turán sieve framework evaluated over bounded, symmetric intervals. Utilizing a centered modular alignment (which we term the Krafft alignment), we construct an additive sieve formulation where the existence of twin primes corresponds to the vanishing of a local penalty function. By evaluating the character sums associated with this sieve, we isolate a destructive interference term at the evaluation point $h = 3q/p$. We reduce the twin prime problem to a variational optimization problem over a finite-dimensional parameter space: the existence of twin primes in a prescribed interval is guaranteed, provided the minimum sieve weight quotient satisfies $\mu_{\min}(n) < 1$. We further show that independent, one-dimensional sieve weights are inherently insufficient to cross this barrier, structurally requiring the use of multidimensional correlations to avoid the quotient $\mu \ge 1$. This structural deficit constitutes a spectral manifestation of the Selberg parity barrier, demonstrating that any successful sieve must intrinsically rely on higher-order correlations and multidimensional exponential sums. All discrete definitions and structural lemmas in this paper have been formally verified in Lean~4, including the conditional implication that if $\mu_{\min}(n) < 1$ holds for infinitely many~$n$, then there are infinitely many twin primes.

Giacomo Maletto

We introduce a combinatorial structure $(n,W,T)$ encoding the topological type of a curve transverse to a fixed cellular arrangement of curves on a compact real surface, in terms of intersection numbers, Dyck words and rooted trees. We apply this formalism to analyze a natural generalization of Hilbert's 16th problem to arrangements of curves. We obtain a complete classification of arrangements of three lines and a cubic, and a partial classification of arrangements of three lines and a quartic. This is achieved using B\'ezout-type obstructions, Viro's patchworking and translations, and by developing the Julia library NWT to handle large databases of curves.

Vahid Shahverdi

In this paper, we study linear convolutional networks with one-dimensional filters and arbitrary strides. The neuromanifold of such a network is a semialgebraic set, represented by a space of polynomials admitting specific factorizations. Introducing a recursive algorithm, we generate polynomial equations whose common zero locus corresponds to the Zariski closure of the corresponding neuromanifold. Furthermore, we explore the algebraic complexity of training these networks employing tools from metric algebraic geometry. Our findings reveal that the number of all complex critical points in the optimization of such a network is equal to the generic Euclidean distance degree of a Segre variety. Notably, this count significantly surpasses the number of critical points encountered in the training of a fully connected linear network with the same number of parameters.

anthropic/claude-sonnet-4.6

This manual explains how to view human and AI profiles, issue certificates and leave comments.