Alonso S. Castellanos, Guilherme Tizziotti, Wilson Olaya-León
原文摘要
In this work, we investigate permutation decoding for algebraic geometry (AG) codes arising from algebraic curves defined by separated polynomials. Using automorphisms of the underlying curves, we construct permutation automorphisms of the associated algebraic geometry codes and exploit the resulting orbit structure to determine information and check positions. We introduce a class of curves, called SAP curves (Separated Additive Polynomial curves), and investigate one-point AG codes defined on them. For these codes, we obtain permutation decoding sets that correct burst errors supported on coordinates associated with rational points sharing a common coordinate. We further identify a subclass of special SAP curves, including Hermitian curves, generalized Hermitian curves, and certain maximal curves, for which additional automorphisms yield more powerful decoding sets.
We study how a smooth irreducible variety $X$ of dimension $n$ and degree $d$ in $\mathbb{P}^m$, $m\geq n+2$, can be recovered from the equations of its images under two generic projections from unknown centers. This extends earlier results on curves in $\mathbb{P}^3$ and surfaces in $\mathbb{P}^4$ to any dimension. Since $X$ is the normalization of each image, one image determines $X$ modulo the projective group when $X$ is linearly normal; what two images add is the configuration, that is the pair of projections. For complete intersections we prove unconditionally that the set of configurations compatible with the two images and their Hilbert polynomial is finite, of size at most $|\mathrm{Lin}(X)|$, the projective automorphism group of $X$. The argument is effective and the bound is attained. We then introduce an equation generalizing Kruppa's equation, relating the duals of the images to a fundamental matrix, encoding the pair of projections modulo the projective group. We give a necessary condition for its solution variety to be finite, a bound on its isolated points, and a sufficient condition under which the true fundamental matrix is rigid: no quadric may contain a generic line $L$ together with the locus of points of $X$ whose tangent space meets $L$. Rigidity is unconditional for curves; in higher dimension it holds under a first-order liftability hypothesis, which we certify for a del Pezzo quartic surface in $\mathbb{P}^4$. Global finiteness of that solution variety remains open, necessarily so, as it contains solutions realized by no variety; for $\dim X<\mathrm{codim}\,X-1$ the obstruction amounts to knowing the fundamental matrix itself. Once the projections are known, $X$ is recovered from the two cones over the images: in codimension two their intersection is $X$ plus a component of degree $d(d-1)$; in higher codimension $X$ is its unique component of maximal dimension.
We construct asymptotically good codes by evaluating Siegel modular forms at superspecial points on congruence towers of Siegel modular varieties. For every genus $g\ge2$ and every fixed weight $w\ge g+1$, the construction gives such a family in sufficiently large characteristic, over a finite field which is independent of the level of the tower. The towers can be chosen so that this field is $\mathbb{F}_{p^2}$. We give an explicit lower bound for the asymptotic rate. The lower bound for the relative minimum distance follows from the Pieri formula on the space of flags in the Hodge bundle and an intersection argument along its strata. We also prove that, for $p\ge5$ and $0\le w\le p+1$, the kernel of evaluation in every genus consists precisely of the multiples of the ordinary Hasse invariant. For $g=2,3$, the geometry of the supersingular locus gives explicit bounds for the relative minimum distance and recovery sets whose sizes are bounded independently of the level.
We determine the maximum number of common zeros in projective space of any set of linearly independent homogeneous polynomials of the same degree, when the degree is smaller than the cardinality of the finite field. This proves the conjecture of Beelen, Datta, and Ghorpade, resolving a longstanding problem on common zeros and determining the complete weight hierarchy of the corresponding projective Reed-Muller codes. The proof uses Green's hyperplane restriction bound, combined with a footprint argument that reduces the problem to the affine case solved by Heijnen and Pellikaan. Wei duality then determines the hierarchy in sufficiently large degrees, recovering the complete hierarchy in every degree for the projective plane. Finally, we prove an upper bound valid in every degree, and conjecture that it is always sharp.
We prove that for every open $U \subseteq \mathbb{R}^n$ the spherical projection of every gradient trajectory $θ(t)$, $t \geq 0$, of a definable $C^1$-potential $F \colon U \to \mathbb{R}$ has finite length and, in particular, converges (directional convergence). In other words, every gradient trajectory of a definable $C^1$-potential has finite total angular motion. Here, definable means definable in an arbitrary o-minimal structure expanding the real field. Notably, the gradient trajectory in this result is an arbitrary solution of the gradient flow differential equation (whose solutions may not be unique) with no further assumptions: The gradient trajectory may be unbounded and diverge to infinity or it may be convergent. The established theorem therefore provides affirmative answers to both the gradient conjecture at infinity for definable potentials and the definable Thom gradient conjecture. We extend our conclusions from the Euclidean metric to a general class of ambient metrics including Riemannian metrics and we establish uniform local upper bounds for the lengths of the spherical projections of gradient flow trajectories. We finally apply our findings to the training of deep neural networks (DNNs) to conclude that, in the case of definable and continuously differentiable activation and loss functions, the spherical projection of the DNN training flow must always converge (directional convergence).
A fixed finite binary Ising model has only finitely many admissible complex belief propagation fixed points, even at degenerate parameter values. Families of positive fixed points obtained by varying external fields can nevertheless have nontrivial topology. We study this distinction through the external-field projection. For a fixed field, we prove finiteness by bounding the pole orders of the magnetizations on a hypothetical complex solution curve. The generic degree on the support graph then bounds the total local multiplicity. A graph has a single smooth real line as its full positive response for every choice of positive couplings, base field, and strictly positive sweep direction exactly when it contains no two vertex-disjoint connected subgraphs of cycle rank at least two. Two such subgraphs suffice to create a circular component. For a plane obtained by varying two vertex fields, connected simple bipartite cubic graphs with one uniform coupling and a universal bound on the fixed fields realize every genus. The minimum number of vertices grows logarithmically in the genus, by a matching algebraic upper bound on Betti numbers. Finally, we certify the complete field discriminant of an eighteen-vertex product model and recover its torus component's genus from the counts of distinct roots on all field cells. The three main theorems, including the logarithmic size bounds, are formally verified in Lean 4 with Mathlib.
A collection of classical codes $C_1,\dots,C_m$ has product expansion if every low-weight word in the sum of their extensions to the product space can be written as a sum of words supported on few lines, each line being a codeword of the corresponding $C_i$. For $m=2$ product expansion is equivalent to robust testability of the tensor code, while for $m\ge 3$ it is strictly stronger; it is the hypothesis on the local codes behind recent constructions of good quantum LDPC codes, good quantum locally testable codes, and quantum codes with transversal non-Clifford gates. No explicit family over a fixed field was previously known to achieve constant rate, linear distance for both the code and its dual, and 3-fold product expansion at growing length. We prove that for the one-point evaluation codes $C_i$ of length $N_i=728\cdot27^{\,i-1}$ and pole order $\lfloor N_i/5\rfloor$ on the Garcia--Stichtenoth tower over $\mathbb{F}_{729}$, the triple $(C_i,C_i,C_i)$ has product expansion with a constant independent of $i$. The dual codes $C_i^\perp$ then form an explicit fixed-field family of rate tending to $109/130$ and relative distance at least $8/65$. The proof is geometric: we view the words of the sum code as arrays on the product of three copies of the $i$-th curve of the tower, where the parity checks are sections of a line bundle. An exact generation theorem describes the sum-code words supported on the grid footprint of a suitable divisor. A support-localisation argument then shows that every low-weight array lies on such a divisor, after deleting a small number of coordinates. We also give a sufficient criterion for three-fold product expansion in terms of degree and genus ratios, yielding explicit families over every non-prime finite field of order at least $512$. The principal conclusions have been formally verified in Lean 4.
Minimal Filling Architectures for Polynomial Neural Networks
Kevin Dao, Jose Israel Rodriguez
AI 速读 基于摘要
该文构造反例推翻“每个最小填充架构的隐藏层宽度都单峰”的猜想，并证明深度不超过三时单峰性成立。
cs.LG · math.AG
查看原文摘要
Minimal filling architectures describe the smallest hidden-layer widths for which the neurovariety of a polynomial neural network fills its ambient function space. It was conjectured that every such architecture has unimodal hidden widths. We construct counterexamples to this conjecture using a frontier search and certify the main example through finite-field Jacobian computations and recursive dimension bounds. In the opposite direction, we prove that every minimal filling architecture of depth at most three is unimodal. A nonunimodal minimal filling architecture of depth four shows that the depth bound is sharp.
We give a machine-checked proof, in Lean 4, of the universal property of multi- centered blowups: given a finite family Z of closed subschemes of a scheme X, presented on each chart by ideals, there is a scheme BlZ X over X, terminal among X-schemes on which every member of the family becomes an effective Cartier divisor. To the best of our knowledge this is the first formalisation of blowups, single- or multicentered, in any theorem prover. The proof is organised around the universal property of the multicentered dilatation of a ring. The comparison isomorphisms on overlaps, the cocycle identity, and the independence of the chosen presentation are obtained from the uniqueness part of this universal property. We give the exact theorem statement as formalized, the proof architecture, and the main new mathematical ingredients, together with the corresponding Lean code.
We study the least pole order of a rational function whose nonzero evaluations scale a prescribed code to a Euclidean LCD code. A residue-one differential yields an obstruction for odd-dimensional one-point AG codes in characteristic two: for $C=C_L(D,aQ)$ with $0\le a<n$ and odd dimension, the threshold is at least $n+2g-a-1$. The obstruction uses alternating Gram matrices and does not require the scaled code to be self-orthogonal. For the dimension-three codes $C_L(D,(r+1)Q)$ on Hermitian curves over $\F_{r^2}$, $r=2^e\ge4$, we obtain \[ n+2g-r-2\leτ_D(C;Q)\le n+2g-3. \] Consequently the leading genus contribution is asymptotically sharp even in dimension three. On the elliptic curve $y^2+y=x^3$ over $\F_4$, we prove the exact dimension-three threshold $6$ by an explicit multiplier. For repetition codes on the Hermitian family, the threshold equals the general interpolation bound $n+2g-1$. The upper bounds use information sets and hold for $q>3k$, and for $q>2k$ in characteristic two.
We construct $46{,}992$ new families of Fano $4$-fold hypersurfaces $X_d\subset\mathbb(a_1,\ldots,a_6)$ of Fano index $1$ with $d>200$, whose general members have terminal singularities. Unlike the $11{,}617$ quasismooth families classified by Brown and Kasprzyk, these are nonquasismooth, for which determining terminality can require the analysis of hyperquotient singularities. An exhaustive search has classified all nonquasismooth families with $d\leq200$, but extending this classification to higher degrees becomes computationally prohibitive. We therefore exploit the sparsity and spatial clustering of terminal examples using two best-first searches over $\mathbb{Z}^6$: one with a fixed heuristic, and the other with a neural network trained online by deep reinforcement learning. Together, the searches reach degrees up to $3{,}234$. The neural network additionally discovers an average of $79$ hard-to-reach families per run. For the most isolated examples, our conservative distance-based model gives an estimated lower bound of nearly $40\%$ of the fixed search's total step budget to discover a single such family. An ablation study demonstrates that this advantage arises from the neural network rather than stochastic exploration alone. The complete list of families is publicly available.
We study nearest and farthest Voronoi diagrams of lines in three-dimensional Euclidean space, which partition space according to the closest and farthest input lines, respectively. For arbitrary lines, a gap remains between the known quadratic lower bound and near-cubic upper bounds for the complexity of the nearest diagram. We consider input lines chosen from one of two continuous families of straight lines on a smooth surface defined by a quadratic equation, with one line from each family passing through every point of the surface. Under the geometric assumptions stated in the paper, we prove quadratic upper bounds on the total number of vertices, edges, faces, and connected three-dimensional regions in both diagrams. We also construct quadratically many nearest vertices using lines from a single family on one fixed surface, establishing a tight quadratic worst-case bound. The farthest diagram has quadratic total complexity for every input satisfying our assumptions. Our proof uses a parametrization that arranges the input lines in a cyclic order and expresses tangency to a sphere by a polynomial equation of degree four. At a Voronoi vertex, the four tangent input lines give four distinct roots. The sign changes between these roots, together with the nearest or farthest condition, force two intervening parameter intervals to contain no other input parameters. Thus, only quadratically many choices of four lines can define a vertex, and each choice yields at most eight isolated candidate centers.
Cicero Carvalho, Maria Vaz Pinto, Rafael H. Villarreal
原文摘要
Let $X$ be a projective nested product of fields and let $δ_X(d)$ be the minimum distance in degree $d\geq 1$ of the projective nested Cartesian code $C_X(d)$. The regularity index ${\rm reg}(δ_X)$ of the minimum distance function $δ_X$ is the minimum integer $d_0\geq 0$ such that $δ_X(d)=1$ for $d\geq d_0$. We give a formula for ${\rm reg}(δ_X)$ by determining an indicator function of least degree for each point of $X$ and using the fact that ${\rm reg}(δ_X)$ is the ${\rm v}$-number of the vanishing ideal $I_X$ of $X$. Then we give an arithmetical criterion that characterizes when $X$ is Cayley--Bacharach.
Every real spherical harmonic of even degree is congruent to its mirror image. We prove that a real spherical harmonic of odd degree $n\ge3$ with distinct Maxwell axes is congruent to its mirror image if and only if the hyperelliptic curve of genus $n-1$ defined by its binary form has a reduced automorphism of even order whose lift acts on the form by $-1$. We then determine the locus of these curves: it is the locus of curves with an extra involution if and only if $n-1$ is a power of two, and otherwise it also contains, for each even divisor $k$ of $n-1$ with odd cofactor $d\ge3$, an irreducible family $y^2=x K(x^{2k})$, $°K=d$, with generic reduced automorphism group $\Z/2k$. For $n=3$ the condition is the vanishing of the skew invariant of the binary sextic. As an application, the invariants of the truncated spherical harmonic expansion of a genus-zero surface, in a conformal parameterization normalized by its conformal barycenter, give a descriptor that is exactly invariant under parameterization, position and scale, records handedness, and is stable in the orbit distance.
For every fixed genus $g\geq2$ and every $0<ρ<1$, we construct maximum distance separable (MDS) algebraic geometry codes over $\F_q$ for all sufficiently large odd squares $q$. Their parameters are $\bigl[n,\lfloorρn\rfloor,n-\lfloorρn\rfloor+1\bigr]_q$ with $n=\sqrt q+1+O_g(q^{1/4})$. The codes use full Riemann--Roch spaces on genus-$g$ hyperelliptic curves. They are not monomially equivalent to generalized Reed--Solomon (GRS) codes when $q$ is sufficiently large. An infinite subfamily has exact length $n=\sqrt q+2gq^{1/4}+1$.
We study the maximal length of MDS algebraic geometry codes from genus-two curves over finite fields. Let $C_\mathcal{ L}(D,G)$ be such a code over $\mathbb F_q$, with length $n$ and dimension $k$. Assume that $3\leq k\leq n-3$. We prove that $n<(2q+10\sqrt q+16)/3$. This bound holds for every prime power $q$ and for both parities of $k$. It gives a uniform upper bound of $2q/3+O(\sqrt q)$ for the maximal length. The explicit bound is strictly smaller than $q+1$ when $q\geq125$.
How many fixed points does belief propagation have on a binary pairwise model? Over complex parameters, the admissible equations of a finite loopless multigraph form an algebraic family whose generic degree we determine: it is a graph state sum over edge sets without vertices of degree one, each vertex of positive degree contributing one less than its degree. We give two geometric realizations of this state sum. The first is the total Betti number of a universal edge-incidence ring constructed without parameters. Its full graded Betti table is the state sum, its Tor algebra is natural for noncontracting graph maps, the alternating numerator recovers the graph polynomial of Watanabe and Fukumizu, and an Artin reduction by a regular sequence realizes the one-variable specialization, corrected for tree components. The second lies in the actual family: at the strong-field boundary each state factor is a ramification index, and at sufficiently strong field the state charts exhaust the admissible complex fiber. For fixed nonzero real couplings, every Eulerian message-sign sector contains a real fixed point at every external field, and strong fields attain this lower bound: one admissible real fixed point per element of the binary cycle space and one positive fixed point. For graphs of maximum degree at most three, this minimum equals the number of minimal module generators of the edge-incidence ring. One state decomposition connects the complex degree, graded syzygies, and an attained minimum real count. The degree, selected ring and resolution statements, and the real counts are formalized in Lean 4.