Papers

In this second paper on descent methods as proof mechanisms in harmonic analysis (the first treated Bourgain's $\Lambda(p)$ selection theorem), entropic mirror descent gives an improved restricted isometry bound, improving sampling and recovery bounds in the Fourier Ratio program. Let $H$ be a complex $n\times n$ matrix with $H^*H=nI$ and $|H_{ij}|=1$. For fixed $0<\varepsilon<\frac14$ and $A_0>0$, independent Bernoulli row selection with expected cardinality $C_{\varepsilon,A_0}r\log\frac{2en}{r}\log(2r)\leq m\leq n/2$ preserves, after normalization by $\sqrt m$, the squared norm of every complex vector $y$ with $\|y\|_1\leq\sqrt r\|y\|_2$ within a factor $1\pm\varepsilon$, with failure probability at most $Cn^{-A_0}$. This includes every $r$-sparse vector and also fully supported ones; support size enters only through this inequality. For $y=\widehat f$ the condition is exactly $\operatorname{FR}(f)^2\leq r$, where $\operatorname{FR}(f)=\|\widehat f\|_1/\|\widehat f\|_2$, giving uniform energy sampling and approximate recovery even for signals with full Fourier support. The same holds for a uniform subset of prescribed cardinality, and for discrete Fourier and real Hadamard matrices. At fixed accuracy the bound removes one sparsity logarithm from the earlier count and replaces $\log n$ by $\log\frac{2en}{r}$; for Walsh matrices it matches, up to constants, the lower bound of Błasiok et al.\ in their range. One relative-entropy potential controls the corrections of an amplitude predictor at every scale; counting them on the symmetric difference of two samples gives the uniform estimate. We also prove the sufficient bound $C_{A_0}\varepsilon^{-5}r\log\frac{2en}{r}\log\frac{2r}{\varepsilon}$, and stable sparse recovery from $Cs\log\frac{2en}{s}\log(2s)$ measurements, with error controlled by the noise and the best $s$-term approximation.

The physical deformation complex of heterotic $G_2$ compactifications couples instanton deformations to geometric fields and their first-order $\alpha'$ corrections. We give an explicit chain-level decoupling of its first-order coefficient complex at the minimal standard embedding, where the gauge bundle is the tangent bundle, the gauge connection is the Levi–Civita connection, and the background $G_2$ structure is torsion-free. The first Atiyah coupling is null-homotopic through a skew covariant derivative extending the induced connection variation. A second homotopy, given by the covariant codifferential of the skew part of an endomorphism-valued cochain, removes the coupling from the bundle sector to the geometric sector. The residual geometric curvature term cancels against the contribution of the first transformation. These identities hold on arbitrary cochains in every canonical degree and yield mutually inverse differential-operator chain maps. Consequently, the cohomology decomposes into instanton and geometric coefficient cohomologies in all degrees. The argument requires only $\mathrm{Hol}(g)\subseteq G_2$. To our knowledge, no previous explicit all-degree splitting removes the internal gauge and geometric couplings of this physically reduced standard-embedding coefficient complex.

Graph theory is a powerful tool in quantum computing, with applications ranging from quantum circuit synthesis and optimization to entanglement mapping. Recent work has shown how one can use edge-colored graphs to model photonic experiments that generate GHZ and W states. However, the latter work also proved that verifying that a graph models a Dicke state experiment is coNP-complete. In this work, we provide families of graphs that generate $|D_{k}^a\rangle \otimes |0\rangle^{\otimes b }$, where $b = |a-2k|$ is the number of spectator modes. The graph setup consists of a doubled complete subgraph on $a$ vertices and a collection of auxiliary vertices. We prove that every coincidence carries exactly $k$ excitations, every weight-$k$ computational basis state on bitstrings of length $a$ is realized, and each of those bitstrings is realized exactly $(n/2)!$ times, where $n = a+b$. Since verifying the Dicke FORALL condition is coNP-complete in general, constructing explicit families that provably generate Dicke states is of interest.

This paper is concerned with the Fleming-Viot particle system introduced by Burdzy, Hołyst and March (2000). In this model, $N$ Brownian particles evolve independently in a bounded domain $D$ until one of the particles reaches the boundary. Then, the particle which has hit the boundary instantaneously jumps to the location of one of the other particles, chosen uniformly at random. Burdzy, Hołyst and March showed that when $N$ tends to infinity, the empirical measure of the system converges to a solution to the heat equation on $D$ with Dirichlet boundary conditions, renormalized to have total mass $1$. Our main result is a sharp, uniform-in-time, quantitative version of this result. We employ the method of weak propagation of chaos, which necessitates a careful study of the (backward) Kolmogorov equations associated to the $N$-particle systems, and the infinite-dimensional transport equation whose characteristics are given by renormalized solutions of the Dirichlet heat equation. The proofs are entirely analytical, and most of the technical effort is devoted to building barrier functions which are used to control the singular behavior of the system when most of the particles approach the boundary.

We find all ring of integers, R, and finite groups G, such that RG has almost trivial units. Moreover, for such a group ring we determine the Swan subgroup of its locally free classgroup

We construct explicit families of maximal singular curves over finite fields. A maximal singular curve is a curve that attains the Aubry-Perret bound, which is the natural extension of the Hasse-Weil-Serre bound on the maximum number of rational points on a smooth curve over a finite field. Starting from a smooth elliptic or hyperelliptic curve $\tilde{C}$ over $\mathbb{F}_{q}$ ($q$ odd), we generate singular curves $C$ with non-split nodes or cusps. In our approach we use linear projections of geometric embeddings, and we apply Stöhr's embedding of hyperelliptic Gorenstein curves and the Rosa-Stöhr theory of trigonal Gorenstein curves. The construction is applied to the Tafazolian and Tafazolian-Torres smooth maximal curves. Finally, we also determine the gonality of all obtained curves.

The category Nom of nominal sets was proposed by Pitts and Gabbay as a setting for the semantics of abstract syntax with variable bindings. It is well-known that Nom is a topos, also known as the Schanuel topos, and in particular it follows that Nom is locally closed, i.e., every slice category Nom/X is cartesian-closed. In this paper, we show that Nom has a much stronger property: every monoidal (closed) structure on Nom induces a monoidal (closed) structure on all slice categories Nom/X. One monoidal closed structure of particular interest on Nom is the separated product A * B, whose right adjoint A -* B is called the separated function space. In particular, it follows that Nom is locally separatedly closed.

We establish the positive semidefiniteness of a class of Hermitian Cauchy-like matrices associated with real Hurwitz polynomials having distinct zeros. Writing the zeros as $-\lambda_1,\ldots,-\lambda_n$, the matrix entries are defined through ratios of elementary symmetric polynomials in the variables $\lambda_i$. The result extends an earlier positivity theorem obtained under the assumption that all zeros are real. We first show that the coefficients appearing in the denominators are nonzero, so that the matrices are well defined. We then construct an explicit congruence between each matrix and the solution of a Lyapunov equation whose state matrix is in companion form. Positive semidefiniteness follows from a recent result on such equations with entrywise nonnegative right-hand-side data. This approach establishes the extension to complex zeros through the connection between structured matrices and Lyapunov equations.

We study genus two curves whose Jacobians have complex multiplication by a biquadratic CM field. In this setting the Jacobian decomposes as a product of two elliptic curves with complex multiplication by orders in the same imaginary quadratic field. We derive upper bounds for the primes of stable bad reduction of such curves in terms of the discriminants of the CM orders. To do so, we give an explicit description of principally polarized abelian surfaces with CM by a biquadratic field via small CM cycles on an orthogonal Shimura variety of signature (3,2). We use it to prove an arithmetic intersection formula between such CM cycles with Humbert surfaces in terms of coefficients of incoherent Eisenstein series and representation numbers of ternary positive definite quadratic forms. Finally, employing the arithmetic properties of higher Green functions, we show that primes of stable bad reduction always exist.

We establish several sufficient conditions ensuring that a perturbed family of operators in a given Hilbert-Schmidt frame for a separable Hilbert space remains a Hilbert-Schmidt frame. The perturbation criteria are formulated through Nagy-type and Pollard-Hilding-type inequalities, yielding explicit estimates for the resulting frame bounds. Several examples are presented to illustrate the applicability of the results.

We study frames of unilateral iterations generated by finitely many vectors under an invertible bounded operator $T$. Suppose that the spectrum of $T$ lies in $\{\rho e^{it}:0<\rho\le1,\ |t|\le c\}$, where $c<\pi$, and that its essential spectrum lies in $\{e^{it}:|t|\le c_{\mathrm e}\}$, where $0\le c_{\mathrm e}\le c$. Real powers are defined through the principal logarithm. Our structural result identifies fractional-orbit synthesis, up to an invertible map and a compact perturbation, with synthesis of projected vector-valued exponentials on an arc containing the essential spectrum. Semi-Fredholm properties and Fredholm index therefore transfer. For a uniformly separated temporal set $\Lambda$, this reduction gives a frame when its one-sided lower uniform density satisfies $D_+^-(\Lambda)>c_{\mathrm e}/\pi$ and its logarithmic block density satisfies $L(\Lambda)>c/\pi$. Completeness at the latter threshold requires only an invertible operator with spectrum in the corresponding angular sector and a complete integer multiorbit. At $L(\Lambda)>2c/\pi$, the complement property is equivalent to a finite condition in channel-cyclic subspaces, with real phase-retrieval consequences. Generalized divided differences admit the same compact reduction and the analogous frame theorem with multiplicities counted. For arbitrary one-point-per-cell sampling with cell length $r$, the sufficient raw-sampling conditions are $rc<\pi$ and $2rc_{\mathrm e}<\pi$. Thus, the full range $rc<\pi$ holds for raw samples when the essential spectrum is contained in $\{1\}$; over the unrestricted class, the sharp universal range is $rc<\pi/2$. Divided-difference preconditioning restores $rc<\pi$ for all operators considered and converts exact collisions into logarithmic-derivative data.

In this paper, we consider diffusion processes on a graph. On the $i$-th edge, each process is governed by an operator of the form $D_{v_i}D^+_{u_i}$, and the behavior at the vertices is determined by the gluing conditions. The paper characterizes the weak convergence of a family of such processes to a limiting process in terms of the behavior of the scale and speed functions $u_i^n$ and $v_i^n$ and the parameters of the gluing conditions. This may be viewed as a generalization of Freidlin and Wentzell's result on the weak convergence of a family of processes on a line.

We define a natural orientation on a spanning-tree-decorated planar map whereby, roughly speaking, each directed edge in the map is oriented to match the direction of the contour exploration of the spanning tree. We study directed distances (lengths of shortest directed paths) with respect to this orientation. We construct the Busemann function which measures directed distances to $\infty$ along a natural interface in the uniform infinite spanning-tree-decorated map. We show that this Busemann function, re-scaled appropriately, converges in law to a $3/2$-stable Lévy process. We also show that in a uniform spanning-tree-decorated map with $n$ edges, directed distances are typically of order $n^{1/3}$. Using a strong coupling argument, we deduce analogous statements for directed distances in other random planar maps in the $\sqrt 2$-Liouville quantum gravity (LQG) universality class, including uniform meandric systems and mated-CRT maps for $\gamma=\sqrt 2$. These results give the scaling dimension for a hypothetical directed version of the $\sqrt 2$-LQG metric. Our proof strategy is inspired by work of Borga and Gwynne (2025) on directed distances in bipolar-oriented triangulations.

An investigation of local aspects of the space of axisymmetric traveling-wave solutions to the three-dimensional Euler equations near Hill's zero-flux spherical vortex and Norbury's small-flux vortex rings is the endeavor undertaken herein. The aforementioned family consists of idealized, swirl-free, and nonsmooth objects, their azimuthal relative vorticity being prescribed by a scaled characteristic function of the compact vortex core. We confront the desingularization question, both within the swirl-free solution subspace and upon admitting swirl, and prove that each member of the Hill–Norbury family of sufficiently small flux is the limit of traveling-wave solutions to the Euler equations that are smooth to infinite order and have compact toroidal vorticity support. In the latter swirling regime, however, a curious flexibility is revealed: total vorticity may diverge along sequences converging in a natural desingularization topology. Our construction, founded upon a five-dimensional reformulation of the Stokes stream function, is an amalgamation of delicate thin-domain analysis, singular-parameter Newton-type nonlinear inversion, and topological fixed-point arguments. To the best of the authors' knowledge, this constitutes the first verification that the small-flux Hill–Norbury family is nowhere isolated from the collection of smooth vortex rings.

We study the proper geometric dimension of full mapping class groups of non-orientable surfaces with punctures. Building on the computation for closed non-orientable surfaces, we prove that the proper geometric dimension agrees with the virtual cohomological dimension in all but a finite collection of low-complexity cases, for which we obtain explicit bounds. Our results provide the non-orientable counterpart of the corresponding computations for mapping class groups of closed and punctured orientable surfaces.

Locally recoverable codes (LRCs) combine global error correction with the ability to repair a small number of erased coordinates by accessing only a limited number of other coordinates. Motivated by their quantum counterparts, we construct several families of optimal cyclic $(r,\delta)$-LRCs that are Euclidean or Hermitian dual-containing. In the Euclidean case, we obtain four families of optimal dual-containing cyclic $(r,\delta)$-LRCs over $\mathbb{F}_q$ with a range of minimum distances extending beyond the local distance $\delta$. In the Hermitian case, we derive analogous families over $\mathbb{F}_{q^2}$ that are Hermitian dual-containing, when $(r+\delta-1)\mid(q^2-1)$. In addition, we develop a distinct construction for the case $(r+\delta-1)\mid(q^2+1)$ using symmetric defining sets and odd $\delta$, which yields optimal codes with minimum distances $\ell+2$, $\delta+2$, $2\delta-2$, and $2\delta$. The Euclidean Calderbank-Shor-Steane (CSS) and Hermitian stabilizer constructions then give corresponding quantum cyclic $(r,\delta)$-LRCs over~$\mathbb{F}_q$. Further, the resulting stabilizer codes are pure; they meet the relevant quantum Singleton-type bound, and are therefore optimal. We also provide a comparison with previous works, highlighting the parameter regimes and minimum distance ranges covered by our results that are not attained by existing constructions. Explicit examples illustrate the constructions and verify the dual-containment and purity conditions.

We develop a unified algebraic approach to spin-boson transformations based on the formalism of $f$-deformed oscillators. In the single-mode case, we show that the Holstein-Primakoff and Dyson-Maleev transformations, together with the interpolating $\alpha$-family, arise as different factorizations of the same algebraically determined object. The standard spin-boson mappings can thus be interpreted as realizations of a common structure, which makes it possible to clearly separate the exact algebraic content on the physical subspace from effects associated with non-Hermiticity, the choice of metric, and extensions beyond the physical subspace. In the two-mode case, the same approach yields both deformed versions of the Jordan-Schwinger transformation and new exact two-mode realizations. Our results provide a unified description of known spin-boson transformations and naturally lead to new bosonic representations.

We establish conditional limit theorems, pointwise in the initial state, for absorbing Markov chains on a compact metric space $M$. We assume $L^1(M,\rho)$-continuous transition densities, irreducibility and aperiodicity. For the observable $f_\beta(x)=d_M(x,x_0)^{-\beta}$, with suitable $x_0$ satisfying $\rho(B_r(x_0))\sim C_d(x_0)r^d$, we prove that the point-process of normalised observations converges to a Poisson random measure. This yields totally right-skewed $\alpha$-stable laws for $\alpha=d/\beta\in(0,2)$ and, at the boundary value $\alpha=2$, a Gaussian limit with the non-standard normalisation $\sqrt{n\log n}$. We also establish a conditional central limit theorem for $L^2$ observables, exponential deviation bounds for bounded observables and a conditional Poisson law for visits to shrinking targets. The results and remarks have been formalised in Lean~4.

We study variation functions of some lacunary series along the sequence of $b$-adic partitions, where $b>1$ is an integer. By such a lacunary series, we mean that in the definition of the well-known Weierstrass function, the Lipschitz continuous cosine and sine functions are replaced by a continuous and piecewise continuously differentiable periodic function whose Fourier series has zero $(jb)^{\mathrm{th}}$-Fourier coefficients for all non-zero integers $j$. In the proofs, we use a probabilistic approach applying a martingale central limit theorem. Our result extends some recent results of Han, Schied and Zhang (2021), and can be applied when one chooses the periodic function in question, for example, as the Takagi function, a Weierstrass function or a certain lacunary series itself.

This study introduces a decision-epoch-based dynamic electric autonomous dial-a-ride problem (Dyn-EADARP), in which incoming requests are collected and processed at periodic decision epochs. A key decision is not only how to serve requests, but also when to dispatch them. At each decision epoch, the service provider decides which requests to serve and which to postpone, while jointly determining vehicle routes, schedules, and charging decisions. Unexecuted parts of existing plans can be revised as new information becomes available. To solve this problem, we develop an ML–CO policy following the combinatorial optimization augmented machine learning (COAML) framework, which combines a statistical model with a combinatorial optimization layer for decision making. The statistical model predicts prizes for available requests, and a prize-collecting E-ADARP uses these prizes to jointly determine request selection, routing, scheduling, and charging. The statistical model is trained directly to improve the decisions produced by the optimization layer. Computational experiments on 320 test instances demonstrate the efficiency of ML–CO, which solves instances with nearly 500 requests in about 6 seconds on average. It achieves 8.7%–14.3% lower objective values than benchmark policies and an average gap of 4.8% to the anticipative reference. The results provide several managerial insights. First, serving requests immediately is not always best, as selectively postponing some requests can create better ride-sharing opportunities. Second, revising existing plans preserves operational flexibility and substantially improves solution quality. Finally, more frequent decision making does not necessarily improve performance, highlighting the importance of choosing an appropriate decision frequency.

We present a candidate counterexample to a conjecture stating that the existence of a Lyapunov-Krasovskii functional with a pointwise dissipation rate is sufficient for the input-to-state stability of time-delay systems. The counterexample has been derived through interactions with large language models.

Let $n,d\geq1$, and suppose that $E\subset \mathbb{Q}_p^n$ contains a degree-$d$ polynomial image of $\mathbb{Z}_p$ in every leading direction of a nonempty set $V\subset\mathbb{P}^{n-1}(\mathbb{Q}_p)$. We prove the sharp bound $\dim_H E\geq\dim_H V+1$. For joint block-valued polynomial evaluations, the total incidence dimension is at least one plus the supremum of the image dimensions. For analytic coefficient sets, Haar-almost every evaluation attains this supremum, and we bound the Hausdorff dimension of the parameters where the image dimension drops. For bounded coefficient sets and arbitrary fixed anisotropic block scales, we prove a quantitative covering comparison with a common Haar-null exceptional set. Outside this set, each evaluation's covering number dominates the maximum over any compact reference ball, up to any positive power loss at all sufficiently small scales. The proofs use the polynomial-to-line lifting map due to Nadjimzadah, together with Dhar's prime-power Kakeya set estimate and a tube maximal inequality derived from it.

Second-order methods for smooth monotone variational inequalities reach the optimal rate $O(T^{-3/2})$, but a distributed exact Jacobian costs $d$ times more communication than an operator value. We show that similarity does part of the work for free: if the server's Jacobian differs from the global one by at most $\beta$, using it gives $O(L_1D^3T^{-3/2}+\beta D^2T^{-1})$ at first-order communication cost. A quasi-Newton approximation of the residual Jacobian $\nabla F-\nabla F_1$, built from secants already communicated, improves the model but cannot remove the $T^{-1}$ term, because any uniform bound on the Jacobian error leaves it in the rate. We therefore certify the surrogate only along the candidate step: one Jacobian-vector product tests it, and a failed test is reused as an exact correction. This attains the exact rate $O(L_1D^3T^{-3/2})$ while transmitting only vectors. Experiments on LIBSVM and synthetic instances measure accuracy against communication.

We prove a criteria for the Behrend function values on the Hilbert scheme of $n$-points on $\C^3$ from local monodromy and the Denef–Loeser's Lefschetz trace formula of jet spaces. If the power two, three, or four of the monodromy is unipotent, it reduces respectively to a Hessian parity formula, a cubic-complement formula, or a quartic formula. This gives a Milnor fiber and monodromy based proof of nonconstancy of the Behrend function on $\Hilb^n(\C^3)$ after Jelisiejew-Kool-Schmiermann. We also calculate two ideals in $\Hilb^9(\C^3)$ and $\Hilb^{12}(\C^3)$ so that the Behrend function is not $\pm 1$, answering a question of Ricolfi.

Let \(\lambda\succ\gamma\succ\mu\) be equal-degree exponent partitions with at most three parts, and let \begin{equation*} P_{\lambda,\gamma,\mu} = J_\lambda+J_\mu-2J_\gamma, \end{equation*} where \(J_\nu\) denotes the symmetric monomial orbit sum associated with \(\nu\). We prove a necessary and sufficient collision criterion for the positivity of this three-point majorization difference in three variables. For nonnegative integer exponent partitions, \begin{equation*} P_{\lambda,\gamma,\mu}(x,y,z)\ge0 \qquad(x,y,z>0) \end{equation*} if and only if \begin{equation*} P_{\lambda,\gamma,\mu}(t,1,1)\ge0 \qquad(t>0). \end{equation*} Thus the positivity of a genuinely three-variable symmetric polynomial of this form is completely determined by its one-variable restriction to the locus where two variables coincide. The theorem gives a uniform and effectively checkable criterion for an infinite class of symmetric polynomial inequalities. For a fixed integer chain, the global three-variable problem is reduced to a single univariate polynomial inequality, which can often be verified exactly by factorization, Sturm's theorem, or other one-variable methods. The criterion extends well beyond the classical Schur family, produces explicit inequalities outside the Schur pattern, and yields genuine strengthenings and refinements of Schur's inequality.

We use the results of Lorist and Schwenninger, A solution to Crouzeix's conjecture, arXiv:2608.03841v2, https://arxiv.org/abs/2608.03841, to update the 2019 paper of Crouzeix and Greenbaum, Spectral sets: Numerical range and beyond, SIAM J.~Matrix Anal.~Appl., 40(3):1087-1101. For all regions considered in the 2019 paper, we are able to show that the region is a $K$-spectral set with a bound on $K$ that is smaller than that established in the 2019 paper. In particular, we show that various annular regions are $2$-spectral sets and that a more general convex region with a circular hole or cutout is a $4$-spectral set.

We develop a differential-geometric framework for risk-optimal portfolio reallocation. A long-only portfolio is represented as a point of the probability simplex, endowed with a positive-definite Riemannian metric combining market covariance risk with position-dependent concentration risk. The cumulative risk of a rebalancing trajectory is identified with its Riemannian length, so the least-risk transition between a current allocation and a Markowitz target is a geodesic. We compare this route with direct linear rebalancing and projected gradient ascent. When only market covariance risk is priced, the metric is constant and flat, and the geodesic is exactly the straight-line path. Once position-dependent risk is introduced, the geometry becomes curved and the geodesic weakly dominates competing paths with the same endpoints. A Fisher-Rao concentration term produces modest but systematic savings, while an endogenous crowding metric creates non-convex risk ridges that geodesics can bypass through temporary diversification. Numerical experiments, Monte Carlo transitions, and regression analysis show that the largest gains occur when the direct path crosses strongly crowded regions. The framework provides a general geometric formulation of transition management and can accommodate richer risk metrics and transaction-cost structures.

We study the critical behavior of the inhomogeneous semilinear heat equation \[ u_t-\Delta u=|u|^p+w(x) \qquad\text{in }(0,\infty)\times\mathbb{R}^N, \] where \(N\geq3\), \(p>1\), and \(w\in L^1(\mathbb{R}^N)\) is nontrivial and has zero total mass: \[ \int_{\mathbb{R}^N}w(x)\,dx=0. \] This zero-mass case was posed as an open problem by Bandle, Levine, and Zhang in 2000 and has remained unresolved since then. We provide a complete answer to this problem by proving nonexistence of global weak solutions for \[ 1<p\leq\frac{N}{N-2}. \] Combined with the known supercritical existence result for sufficiently small data, this shows that the critical exponent separating the nonexistence and existence regimes remains \[ p_c=\frac{N}{N-2}, \] the same as in the positive-mass case.

We prove an exact isoperimetric inequality for cubes and tori under the $\ell_{\infty} metric. As a corollary, we recover the celebrated grid edge-isoperimetric inequality of Bollobás-Leader.

We study the Bourbaki degree of line arrangements in the projective plane and its interaction with the intersection lattice and the syzygies of the gradient ideal. We show that the Bourbaki degree is obtained by evaluating the reduced characteristic polynomial at the initial degree of the first syzygy module. This leads to a sharp upper bound and to refinements involving the two largest intersection multiplicities, with consequences for the global Tjurina number, the freeness defect, and Terao's conjecture, including an improvement of Dimca's numerical criterion when the smaller exponent is at least five. We also establish addition–deletion formulas and show that the defect from the sharp bound is monotone under line addition. Finally, we prove that the Bourbaki degree is determined by the intersection lattice for arrangements with at most eight lines, while examples with isomorphic intersection lattices and different Bourbaki degrees exist for every number of lines at least nine.