arXiv subjects

Karthik Sheshadri

Publications and source records attributed to Karthik Sheshadri.

9 recordsLinked to original sources

arXiv2026

Conormal Lower Bounds for Determinantal Complexity: Fermat Slices, Tight Bounds, and a Ceiling

The determinantal complexity of a polynomial $f$ is the smallest size of a matrix of affine linear forms whose determinant is $f$; the border version only asks for determinants converging to $f$. The polar method bounds these quantities from below by comparing an invariant of the hypersurface $V(f)$, such as the degree of its dual variety, with the number of solutions of a polynomial system built from a determinantal representation of $f$. We determine how far the method reaches. Lower bounds: if $f$ has a linear restriction to a form of degree $d$ in $N$ variables with smooth projective zero set, then the border determinantal complexity of $f$ is at least $(N-1)(d-1)/(4e)$. Setting the off-diagonal host entries to zero turns every homomorphism polynomial $\hom_{F,q}$ with a connected pattern $F$ into a Fermat polynomial, which gives the bound $(q-1)(|E(F)|-1)/(4e)$. For the VBP-complete path family $W_q$ this is tight: the border determinantal complexity of $W_q$ is $Θ(q^3)$. The same argument gives cubic bounds for natural VP- and VNP-complete families, and a bound in terms of the codimension of the singular locus of $V(f)$. A ceiling: the method, together with every refinement in a precisely defined class, cannot prove a lower bound larger than $O(dN)$ for any polynomial of degree $d$ in $N$ variables. The proof computes the solution count exactly, as the degree of the dual of a generic determinantal hypersurface, and shows that every characteristic-cycle invariant of a hypersurface of degree $d$ is at most $8(d-1)^{N-1}+8N$. For the $n\times n$ permanent the ceiling is $O(n^3)$; for the power sum $x_1^N+\dots+x_N^N$ it is $(1+o(1))N^2/(4e)$, matching the known lower bound.

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arXiv2026

A fully Lean-verified quadratic lower bound on the border determinantal complexity of $x_1^n+\cdots+x_n^n$ over $\mathbb{C}$, including the symmetric and exact models

We present the first end-to-end Lean-verified superlinear lower bound on border determinantal complexity. Let $F_n=x_1^n+\cdots+x_n^n$. We prove, over $\mathbb{C}$, that the border determinantal complexity of $F_n$ is at least $(n-1)^2/(4e)$ and its symmetric border determinantal complexity is at least $(n-1)^2/(2e)$; the same bounds therefore hold for exact determinantal complexity. Together with known $O(n^2)$ constructions, both border quantities are $Θ(n^2)$. The proof compares one enumerative invariant on two sides of a degeneration. On the determinant side, the $(n-2)$nd multidegree of the multiplicity-one Gauss graph of any size-$m$ affine-linear determinant is bounded by an explicit multihomogeneous Bézout number; no smoothness, irreducibility, or squarefreeness is assumed. On the target side, a flat limit of these effective Gauss-graph cycles contains the conormal variety of the projective Fermat cone, whose relevant multidegree is the classical dual degree $n(n-1)^{n-2}$. Effectivity gives the inequality in the correct direction, and an $(n-1)$st root yields the quadratic bound. All four bounds are fully verified in Lean 4 with Mathlib. The Lean theorems assume only $n\ge 3$ and a literal determinantal representation or literal border membership, and depend on no axioms beyond Lean's foundational ones. The formal proof follows a localized version of the argument presented here: where the text compares global cycles, it compares local intersection lengths of finite slices of those cycles. Lean sources: https://github.com/karthik202321/power-sum-dc-lean; archived release: https://doi.org/10.5281/zenodo.23274251

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arXiv2026

Two-Cut Coherence of Quintic Forms: Lifting Separations and Second-Derivative Completeness

For a homogeneous polynomial f of degree d, the degree-k restricted strength C_k(f) is the least number of products needed to write f with factor degrees k and d-k. We introduce a two-cut coherence parameter C_{k,l}(f): the least r such that f = sum_{i,j=1}^{r} p_i m_{ij} q_j with deg p_i = k, deg m_{ij} = l-k, and deg q_j = d-l. This requires two degree interfaces to be realized by a single common factorization. We show it equals the minimum common endpoint width of a three-block compressed transfer network, and equivalently the minimum, over all tensor lifts of f through commutative multiplication, of the larger of the two tensor-train endpoint ranks. In particular it lower-bounds homogeneous ABP width. Our main result is an extraction-completeness theorem for quintics at cuts (1,3). Let D(f) be the largest polynomial slice rank C_1 of a second directional derivative of f, and let t = C_3(f). Over an algebraically closed field of characteristic zero, ceil(D(f)/3) <= Cbar_{1,3}(f) <= C_{1,3}(f) <= t*D(f) + 2t^2, where Cbar denotes border complexity. Hence when C_3 is bounded, ordinary and border two-cut coherence are equivalent up to constants to a one-cut obstruction exposed by a second derivative. We also prove a border-stable lifting separation. For coprime nonzero cubics A and B, the quintic L = abA + cdB has ordinary and border local values C_1 = C_3 = 2, while ceil(max{C_1(A), C_1(B)}/3) <= Cbar_{1,3}(L) <= C_1(A) + C_1(B). Taking A to be a Fermat cubic in n variables, for which we show C_1 = ceil(n/2), gives an unbounded gap between separately optimal local interfaces and a common interface, even in border complexity. This refutes any universal bound of the form C_{k,l} <= C_k + C_l.

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arXiv2026

Trellis State Complexity as an Exact Tropical Factorization Rank

Let $C\subseteq\F_2^m$ be a binary linear code and let $[m]=L\sqcup R$ be a bipartition of its coordinates. The \emph{conditional decoding matrix} of $C$ at this cut is the matrix $W$ indexed by $\F_2^{L}\times\F_2^{R}$ whose entry $W(x_L,x_R)$ is the coset-leader weight $d\bigl((x_L,x_R),C\bigr)$, the minimum Hamming distance from the word $(x_L,x_R)$ to the code. We prove that the min-plus factorization rank (Barvinok rank) of $W$, and likewise its tropical rank, equal $2^{s}$ exactly, where $s=\dim C-\dim C_L-\dim C_R$ is the classical state complexity of the minimal trellis of $C$ at the cut. The upper bound is a two-party reading of Viterbi decoding on the minimal trellis; the contribution is the matching lower bound, which holds against arbitrary min-plus factorizations rather than only sequential trellis realizations, and is obtained from an explicit $2^{s}\times 2^{s}$ tropically nonsingular submatrix built from a transversal of codewords. Specializing $C$ to the cut space of a graph identifies $W$ with the conditional ground-state energy of Ising signings (the frustration index), and yields natural graph families whose conditional matrices have min-plus rank exponential in the number of vertices; for these families we also record the contrasting local statement that all bounded-radius views of a signing are switching-trivial, so the exponential rank is carried entirely by non-local structure. We note explicitly that this rank measures representational incompressibility, not computational hardness: planar families attain the same exponential rank while their ground states are computable in polynomial time.

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arXiv2026

Contested Cluster Selectors: Local Ambiguity, Normal Forms, and Backtracking Cost in Random Constraint Satisfaction

We introduce and empirically investigate \emph{contested cluster selectors} (\CCS): variables that are non-backbone, carry information about solution-cluster identity, and are repeatedly but unreliably forced by local propagation during backtracking search. In instrumented \DPLL{} experiments on random 3-\SAT{} near the empirical satisfiability threshold and on near-optimal random \VC{} instances, a small number of such variables accounts for a large fraction of observed backtracking cost. Pinning two or three high-contestedness variables to solution-consistent values reduces backtracking by 70--80\% on the reference instances studied, and a static degree--polarity metric yields a simple $2^k$ enumeration heuristic with a reported $3.7\times$ speedup over baseline \DPLL{} at $n=50$. A polynomial control experiment on random 3-\XORSAT{} sharpens the interpretation. Gaussian elimination exposes the true affine selector coordinates, whereas \DPLL{} churn concentrates on pivot variables chosen in a poor coordinate system. Thus clustering and non-backbone status are not enough: the empirical hardness signal is \emph{local contestation} that remains after available polynomial-time normal forms. We formalize this distinction through safe coordinate exposers and the \emph{unavoidable contested selector cost} (\UCSC). We also prove an ordered single-pass eraser-memory lower bound: any ordered \FERAM{} that recovers a $k$-bit cluster label from a distribution with residual min-entropy $k-η$ using $S$ bits succeeds with probability at most $2^{S+η-k}$. The paper positions \CCS/\UCSC{} as a structural program connecting backdoors, solution-space geometry, low-degree barriers, and Schaefer-style algebraic normal forms. We do not claim a proof of $P\ne NP$; rather, we isolate the normal-form barrier that any such extension would need to overcome.

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arXiv2026

A symmetric determinantal lower bound for diagonal power sums via polar degree

The symmetric determinantal complexity sdc(f) of a polynomial f is the least m such that f = det(M) for an m x m symmetric matrix M of affine-linear forms. We prove, over the complex numbers, that sdc(sum_{i=1}^n x_i^n) >= (1/(2e) - o(1)) n^2. This is a symmetric companion to the author's non-symmetric polar-degree preprint (arXiv:7680505); the method parallels that work, but the proof here is self-contained and redoes the load-bearing local incidence analysis in the symmetric setting. The general theorem: if X = V(f) in P^{N-1} is a smooth degree-d hypersurface, N >= 3, and f = det(A_0 + sum x_i A_i) with all A_i symmetric of size m, then the top polar degree d(d-1)^{N-2} is at most 2^{N-2} C(m, N-1). The proof uses the symmetric rank-one kernel incidence M(z,x) u = 0. At a genuine polar point M has rank m-1, and a symmetric Schur-complement normal form eliminates the unique kernel line scheme-theoretically; on the resulting local graph the lifted conormal forms u^T A_i u are a common unit multiple of the partials d_i f, so the lifted polar equations cut the ordinary polar slice up to units and each genuine lifted polar point is a zero-dimensional isolated solution. Multihomogeneous Bezout on P^N x P^{m-1} then yields the bound 2^{N-2} C(m, N-1). For F_n = sum x_i^n this gives the constant 1/(2e). More generally, for F_{N,d} = sum_{i=1}^N x_i^d the same theorem gives sdc(F_{N,d}) >= (1/(2e) - o_N(1)) N(d-1) as N -> infinity. We give an explicit symmetric representation of F_{N,d} of size 2N(d+1)+1, so the diagonal bounds are non-vacuous and tight up to a constant. The result is for exact symmetric determinantal complexity in characteristic zero; it is not a border statement and not a uniform positive-characteristic theorem.

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arXiv2019

Building a Benchmark Dataset and Classifiers for Sentence-Level Findings in AP Chest X-rays

Chest X-rays are the most common diagnostic exams in emergency rooms and hospitals. There has been a surge of work on automatic interpretation of chest X-rays using deep learning approaches after the availability of large open source chest X-ray dataset from NIH. However, the labels are not sufficiently rich and descriptive for training classification tools. Further, it does not adequately address the findings seen in Chest X-rays taken in anterior-posterior (AP) view which also depict the placement of devices such as central vascular lines and tubes. In this paper, we present a new chest X-ray benchmark database of 73 rich sentence-level descriptors of findings seen in AP chest X-rays. We describe our method of obtaining these findings through a semi-automated ground truth generation process from crowdsourcing of clinician annotations. We also present results of building classifiers for these findings that show that such higher granularity labels can also be learned through the framework of deep learning classifiers.

cs.CV↗

arXiv2018

Framing Matters: Predicting Framing Changes and Legislation from Topic News Patterns

News has traditionally been well researched, with studies ranging from sentiment analysis to event detection and topic tracking. We extend the focus to two surprisingly under-researched aspects of news: \emph{framing} and \emph{predictive utility}. We demonstrate that framing influences public opinion and behavior, and present a simple entropic algorithm to characterize and detect framing changes. We introduce a dataset of news topics with framing changes, harvested from manual surveys in previous research. Our approach achieves an F-measure of $F_1=0.96$ on our data, whereas dynamic topic modeling returns $F_1=0.1$. We also establish that news has \emph{predictive utility}, by showing that legislation in topics of current interest can be foreshadowed and predicted from news patterns.

cs.CY↗

arXiv2018

The Causal Link between News Framing and Legislation

We demonstrate that framing, a subjective aspect of news, is a causal precursor to both significant public perception changes, and to federal legislation. We posit, counter-intuitively, that topic news volume and mean article similarity increase and decrease together. We show that specific features of news, such as publishing volume , are predictive of both sustained public attention, measured by annual Google trend data, and federal legislation. We observe that public attention changes are driven primarily by periods of high news volume and mean similarity, which we call \emph{prenatal periods}. Finally, we demonstrate that framing during prenatal periods may be characterized by high-utility news \emph{keywords}.

cs.CY↗