arXiv subjects

Yuxin Lin

Publications and source records attributed to Yuxin Lin.

At least 19 recordsLinked to original sources

arXiv2026

Dense Cores in the Vicinity of an HII Region

Massive stars strongly influence their surroundings through radiative and mechanical feedback, but its effects on dense gas structures at sub-pc scales remain poorly constrained. We investigate how feedback from a newly formed massive star affects dense cores in the filamentary molecular cloud IRAS 18530+0215. We analyze ALMA Band 6 observations of 1.3 mm dust continuum and DCN, N$_2$D$^+$, and $^{13}$CS line emission, together with VLA K-band continuum and NH$_3$ observations. Dense cores are identified with astrodendro, and their temperatures, masses, velocity dispersions, and virial parameters are derived. The dynamical state of the ultra-compact H II region is examined through energy and pressure estimates. The H II region has a radius of $\sim$0.1 pc and an expansion velocity of $\sim$2.5 km s$^{-1}$, corresponding to a shell dynamical age of $\sim$0.06 Myr. DCN and $^{13}$CS cores are concentrated near the H II region, whereas N$_2$D$^+$ cores preferentially lie farther away. Core temperatures and velocity dispersions decrease with projected distance from the H II region. Virial parameters increase within the inner $\sim$0.3 pc but decline sharply beyond this scale, while core masses show no significant trend with distance. Strong star formation signatures are found at $\sim$0.2 pc, whereas more distant regions still host quiescent, cold dense cores. The compact H II region appears trapped or choked within $\sim$0.1 pc, while its feedback extends to at least $\sim$0.3 pc. Within this region, feedback enhances core velocity dispersions, gas temperatures, and virial parameters, with no evidence that it promotes the formation of more massive dense cores.

astro-ph.GA↗

arXiv2026

A $k$-Dimensional Version of the Largest Intersection Problem

Suppose we have $r$ linearly independent hypersurfaces of degree $d$ in $\mathbb{P}^m$ (or $\mathbb{A}^m$) defined over a finite field $\mathbb{F}_q$, whose intersection is at most $k$-dimensional. What is the largest possible number of $\mathbb{F}_q$-rational points in the intersection? We conjecture an exact formula for this problem in both the projective and affine settings, assuming $q\geq d+1$. The case $k=m-1$ recovers the Beelen-Datta-Ghorpade conjecture [2] and the case $k=0$ recovers the zero-dimensional conjecture in [15]. Another interesting special case of the conjecture is $k=m-r$, which corresponds to the complete intersection of $r$ degree $d$ polynomials. The case $r=1$, $k=m-1$ was proven by Serre in [16] who showed that if $F$ is a degree $d$ homogeneous polynomial, then $|V(F)(\mathbb{F}_q)|\leq dq^{m-1}+\pi_{m-2}(q)$. We prove the case $r=2$ and $k=m-2$, that is, if $F_1, F_2$ are coprime, degree $d$ homogeneous polynomials, then $|V(F_1,F_2)(\mathbb{F}_q)|\leq d^2q^{m-2}+\pi_{m-3}(q)$.

math.AG↗

arXiv2026

Trace the Self-Gravitating Gas Using CO Isotopologues

Recent studies have shown that the star formation rate (SFR) correlates tightly and linearly with the mass of gravitationally bound gas, which can be delineated from the power-law tail of the column-density probability distribution function ($N$-PDF) derived from dust emission observations. This relationship holds across four orders of magnitude within the Milky Way--spanning low-mass to high-mass star-forming regions and encompassing the extreme environment of the Central Molecular Zone. Building on this framework, we present a new approach for estimating the mass of gravitationally bound gas in molecular clouds using multi-line CO isotopologue observations. Our sample includes 16 molecular clouds with robust detections in $^{12}$CO, $^{13}$CO, and C$^{18}$O $J$ = 1-0, spanning both massive inner Galaxy clouds and nearby star-forming regions. We find that the $N$-PDFs derived from combined CO isotopologue data recover the characteristic log-normal plus power-law profiles seen in dust-based studies. The mass and spatial distribution of the self-gravitating structures estimated from both dust-based and CO-based methods agree well throughout the sample. This indicates that the CO isotopologue combination can robustly trace the self-gravitating component via the $N$-PDF method and provides a reliable, scalable, and velocity-resolved alternative to dust emission for identifying the star-forming gas in molecular clouds.

astro-ph.GA↗

arXiv2026

CRAFTS: Collaborative Role-Adaptive Fine-Tuning of LLM Agents for Chemical Process Simulation

Constructing an executable chemical-process model remains manually intensive. Chemical engineers translate underspecified requests into coupled decisions about unit operations, thermodynamics, streams, specifications, degrees of freedom (DoF), initialization, solver repair, and optimization; one error can invalidate the model. CRAFTS mirrors the staged workflow of chemical engineers by decomposing simulation building into bounded subtasks assigned to seven bounded roles, with deterministic IDAES/Pyomo gates between stages. Given a natural-language request, process flowsheet diagram (PFD) evidence, and curated chemical-engineering knowledge, Input Understanding and Intent recover requirements, constraints, and process semantics; visual, topology, and specification specialists translate them into typed simulator contracts; and Debug and Optimization support bounded repair and eligible optimization. Fine-tuning is applied to the three schema-critical visual, topology, and specification roles, while the remaining roles use untuned Qwen. The resulting VisualGraphIR, TopologyIR, SpecIR, BuildPlan, and SolveReport expose unit, port, thermodynamic, numerical, and execution decisions. Compatible constructors, property packages, and runners are attached only after semantic artifacts pass engineering gates. We introduce OpenIDAES-450, a 450-case IDAES process- simulation dataset, and evaluate the complete seven-role LangChain/LangGraph workflow through solve and eligible optimization on its frozen 82-case held-out split. CRAFTS completes the prescribed validation and execution contract for for 91.5% of cases and achieves unit, stream, and directed-connection F1 scores of 0.815, 0.791, and 0.782. These results demonstrate the effectiveness of role specialization, typed intermediate representations, and deterministic engineering gates for reliable automated process-model construction.

cs.AI↗

arXiv2026

Multi-Scale Magnetic Field Observations Reveal how Colliding Flows Trigger Star Formation

Magnetic fields play a crucial yet complex role in star formation, while their connection between large-scale filamentary clouds and small-scale young stellar objects remains poorly understood. We present new continuum polarization observations from the JCMT, ACA, and ALMA that provide continuous magnetic field measurements from approximately 5 pc down to approximately 4000 au, tracing for the first time the evolution of field morphology seamlessly across all key scales within a massive star-forming system. Our polarization maps reveal multiple U-shaped magnetic field structures pointing toward the central protocluster, aligned with accreting filaments from parsec to subparsec scales and converging at the compact center. This morphology suggests an environment of colliding flows that drag magnetic fields and trigger massive protocluster formation. On approximately 4000 au scales, we identify compact U-shaped fields likely guiding the kinematics of streamers accreting onto dense cores. The increasing curvature of these U-shaped patterns is a direct measure of a growing magnetic field tension force, implying a magnetic field strength scaling index of 0.50 +/- 0.10. These results indicate that the field, possibly enhanced by large-scale flow collisions, becomes strong enough to regulate star formation, linking parsec-scale colliding flows, a subparsec hub-filament system, and the triggering of massive star formation.

astro-ph.GA↗

arXiv2026

CoCoVideo: The High-Quality Commercial-Model-Based Contrastive Benchmark for AI-Generated Video Detection

With the rapid advancement of artificial intelligence generated content (AIGC) technologies, video forgery has become increasingly prevalent, posing new challenges to public discourse and societal security. Despite remarkable progress in existing deepfake detection methods, AIGC forgery detection remains challenging, as existing datasets mainly rely on open-source video generation models with quality far below that of commercial AIGC systems. Even datasets containing a few commercial samples often retain visible watermarks, compromising authenticity and hindering model generalization to high-fidelity AIGC videos. To address these issues, we introduce CoCoVideo-26K, a contrastive, commercial-model-based AIGC video dataset covering 13 mainstream commercial generators and providing semantically aligned real-fake video pairs. This dataset enables deeper exploration of the differences between authentic and high-quality synthetic videos and establishes a new benchmark for highly realistic video forgery detection. Building on this dataset, we propose CoCoDetect, a detection framework integrating contrastive learning with confidence-gated multimodal large language model (MLLM) inference. An R3D-18 backbone extracts spatio-temporal representations, while a confidence gate routes uncertain cases to an MLLM for reasoning about physical plausibility and scene consistency. Extensive experiments on CoCoVideo-26K and public benchmarks demonstrate state-of-the-art performance, validating the framework's robustness and generalizability. Our code and dataset are available at https://github.com/DonoToT/CoCoVideo.

cs.CV↗

arXiv2026

Sketch-to-Layout: A Human-Centric Computational Agent for Constraint-Aware Synthesis of Modular Photobioreactors

Building-integrated photobioreactors (PBRs) offer a pathway for carbon-neutral architecture, yet deployment is hindered by configuration complexity and biological maintenance. This paper presents a modular PBR facade system powered by a computational framework reconciling design intent with physical validity. We introduce 'carbon-neutralization bricks' featuring integrated vessel-and-conduit geometry; monolithic fluid channels enable 'plug-and-play' assembly. To navigate the combinatorial complexity of 14 modular geometries, we develop a Computational Sketch-to-Layout Agent that formulates layout synthesis as a Constraint Satisfaction Problem (CSP). Using the CP-SAT engine, the agent treats sparse user sketches as soft priors while enforcing hard constraints like port alignment and global connectivity. This allows non-experts to synthesize fabrication-ready configurations in near real-time. Furthermore, to facilitate autonomous maintenance, we propose a weakly supervised algae health monitoring pipeline. By employing a hybrid CNN-attention backbone and a temporal ranking loss, the system quantifies biological vitality from photographs without absolute ground-truth labels. Experiments demonstrate the CSP solver achieves a 95.5% success rate on grid scales up to 15 x 15. Qualitative evaluations confirm the framework preserves design semantics while ensuring operational integrity. Long-term tests show the vision module produces health trajectories aligned with 14-day biological cycles, suggesting that integrating interactive synthesis with low-cost computer vision can democratize scalable carbon capture systems.

cs.HC↗

arXiv2026

PRODIGE -- envelope to disk with NOEMA VIII. Sulfur oxides trace a shock caused by a streamer in the inner envelope of a protostar

(Abridged) Recently, streamers have been observed causing shocks at the outer edge of protoplanetary disks. The study of sulfur-bearing species can help us to understand the physical and chemical changes caused by infalling streamers toward their landing positions. We study the physical properties traced by SO$_2$ and SO toward the Class I protostar Per-emb 50, which is possibly related to the streamer infalling toward its disk. We present new NOEMA A-array observations as part of the large program "Protostars and Disks: Global Evolution" (PRODIGE). We analyzed the morphology of SO$_2$ and SO, and complement our interpretations with additional H_$2$CO and CO data from the same program. We compared the SO$_2$ and SO morphology with an infalling-rotating model. We applied Bayesian model selection to the brightest SO$_2$ line to disentangle the different kinematic components traced by this molecule. We used Local Thermodynamic Equilibrium (LTE) and non-LTE analyses to determine the temperature and density of the SO$_2$ emission. There are two separate peaks of SO$_2$ emission offset toward the southwest of Per-emb 50, one brighter (peak 1) at about 180 au from the protostar, and a weaker one (peak 2) at about 400 au. Peak 2 is blueshifted with respect to an infalling-rotating envelope. We propose that this peak is caused by the shock between the inner envelope and the streamer. Peak 1 is consistent with the expected envelope motion, and could thus be caused by shocks at the disk-envelope interface, but potential streamer influence cannot be neglected. Both peaks show abundance ratios consistent with a low velocity shock ($\sim 3-4$ \kms) when compared with shock models. Streamers can affect the physical and chemical structure of both disks and envelopes, suggesting that streamers can play an important role in shaping both structures in the embedded stages of star formation.

astro-ph.SR↗

arXiv2026

Knowing When to Ask: Resolving Uncertainty in Human-Robot Joint Planning via Explicit Dialogue and Implicit Intent Cues

Effective human-robot collaboration in open-world environments requires joint planning under uncertainty about the task, the environment, and the human teammate. Communication is the most direct means of resolving such uncertainty, yet most existing systems support only one-way communication: robots listen and act, treating humans as passive supervisors rather than conversational teammates capable of two-way dialogue. We propose a unified human-robot joint planning system in which the robot actively resolves uncertainty through two complementary communication channels. When uncertainty is decision-critical, an uncertainty-mitigation joint planning module engages the human in clarification dialogue: it grounds ambiguous instructions via an LLM-assisted active elicitation mechanism, enumerates traversability hypotheses through a hypothesis-augmented A* search, and computes a cost-optimal querying policy via dynamic programming, so that the robot asks only the questions whose answers actually matter for the plan. When explicit dialogue is unnecessary or impractical, a real-time intent-aware collaboration module instead reads implicit, nonverbal cues, maintaining a probabilistic belief over the human's latent task intent from spatial and directional signals to enable coordination-aware task selection without any communication overhead. We validate the proposed system in both Gazebo simulations and real-world UAV deployments, integrated with a voice dialogue interface and a Vision-Language Model (VLM)-based 3D semantic perception pipeline. Experimental results show that cost-optimal clarification dialogue cuts the interaction cost by 51.9% while maintaining a 100% task success rate, and implicit intent reading reduces the cooperative task execution time by 25.4% compared to the baselines.

cs.RO↗

arXiv2026

Reasoning Knowledge-Gap in Drone Planning via LLM-based Active Elicitation

Human-AI joint planning in Unmanned Aerial Vehicles (UAVs) typically relies on control handover when facing environmental uncertainties, which is often inefficient and cognitively demanding for non-expert operators. To address this, we propose a novel framework that shifts the collaboration paradigm from control takeover to active information elicitation. We introduce the Minimal Information Neuro-Symbolic Tree (MINT), a reasoning mechanism that explicitly structures knowledge gaps regarding obstacles and goals into a queryable format. By leveraging large language models, our system formulates optimal binary queries to resolve specific ambiguities with minimal human interaction. We demonstrate the efficacy of this approach through a comprehensive workflow integrating a vision-language model for perception, voice interfaces, and a low-level UAV control module in both high-fidelity NVIDIA Isaac simulations and real-world deployments. Experimental results show that our method achieves a significant improvement in the success rate for complex search-and-rescue tasks while significantly reducing the frequency of human interaction compared to exhaustive querying baselines.

cs.RO↗

arXiv2026

Learning to Fuse and Reconstruct Multi-View Graphs for Diabetic Retinopathy Grading

Diabetic retinopathy (DR) is one of the leading causes of vision loss worldwide, making early and accurate DR grading critical for timely intervention. Recent clinical practices leverage multi-view fundus images for DR detection with a wide coverage of the field of view (FOV), motivating deep learning methods to explore the potential of multi-view learning for DR grading. However, existing methods often overlook the inter-view correlations when fusing multi-view fundus images, failing to fully exploit the inherent consistency across views originating from the same patient. In this work, we present MVGFDR, an end-to-end Multi-View Graph Fusion framework for DR grading. Different from existing methods that directly fuse visual features from multiple views, MVGFDR is equipped with a novel Multi-View Graph Fusion (MVGF) module to explicitly disentangle the shared and view-specific visual features. Specifically, MVGF comprises three key components: (1) Multi-view Graph Initialization, which constructs visual graphs via residual-guided connections and employs Discrete Cosine Transform (DCT) coefficients as frequency-domain anchors; (2) Multi-view Graph Fusion, which integrates selective nodes across multi-view graphs based on frequency-domain relevance to capture complementary view-specific information; and (3) Masked Cross-view Reconstruction, which leverages masked reconstruction of shared information across views to facilitate view-invariant representation learning. Extensive experimental results on MFIDDR, by far the largest multi-view fundus image dataset, demonstrate the superiority of our proposed approach over existing state-of-the-art approaches in diabetic retinopathy grading.

cs.CV↗

arXiv2025

First mapping of prebiotic molecule CH2NH in a pre-stellar core

We present the first spatially resolved map of methanimine CH2NH in the prestellar core L1544 using the IRAM 30m telescope. The 2$_{0,2}$-1$_{0,1}$ line at 127 GHz was mapped with 20" resolution ($\sim$2800 au), revealing extended CH2NH emission across the core. The peak line intensity coincides with the well-known c-C3H2 peak, while the integrated intensity peaks between the HNCO and dust continuum peaks due to broader linewidths in the latter region. Column densities of CH2NH are $\sim$(0.5-1.4$\times$)10$^{12}$ cm$^{-2}$, corresponding to fractional abundances of $5\times10^{-11}$-$1\times10^{-10}$, with a trend decreasing from the southern, carbon-chain rich region to the dust and HNCO peak in the north. Comparison with complementary molecular maps and the gas-grain chemical model of Sipil\"a et al. suggests that neutral-neutral gas-phase reactions and dissociative recombination dominate in the outer carbon-chain shell. This study demonstrates that CH2NH, a simple nitrogen- and carbon-bearing molecule previously detected with pointed observations in other cold cores, is present and spatially extended in the evolved pre-stellar core L1544. This indicates that prebiotic nitrogen-carbon chemistry continues efficiently up to the onset of gravitational collapse, providing key constraints for astrochemical models and the early stages of chemical complexity leading to amino acids.

astro-ph.GA↗

arXiv2025

RoboDriveVLM: A Novel Benchmark and Baseline towards Robust Vision-Language Models for Autonomous Driving

Current Vision-Language Model (VLM)-based end-to-end autonomous driving systems often leverage large language models to generate driving decisions directly based on their understanding of the current scene. However, such systems introduce multiple risks in real-world driving scenarios. To evaluate whether VLMs are truly viable for autonomous driving, we introduce RoboDriveBench, the first robustness benchmark focused on end-to-end trajectory prediction tasks. This benchmark systematically evaluates two critical categories of real-world challenges for VLM-based end-to-end autonomous driving systems through 11 simulated scenarios encompassing various corruption types, including 6 scenarios of sensor corruption caused by environmental variations, along with 5 cases of prompt corruption resulting from human intervention and data transmission failures. Each corruption type includes 250 unique driving scenarios and 5,689 frames, resulting in 64,559 total trajectory prediction cases per evaluation. To overcome these real-world challenges, we propose a novel VLM-based autonomous driving framework called RoboDriveVLM, which enhances robustness by mapping more multimodal data-e.g., lidar and radar-into a unified latent space. Furthermore, we introduce a new Test-Time Adaptation (TTA) method based on cross-modal knowledge distillation to improve the robustness of VLM-based autonomous driving systems. Through extensive experiments, our work highlights the limitations of current VLM-based end-to-end autonomous driving systems and provides a more reliable solution for real-world deployment. Source code and datasets will be released.

cs.AI↗

arXiv2025

Tails of Gravity: Persistence of Star Formation in the CMZ Environment

We characterize star-forming gas in six molecular clouds (Sgr B1-off, Sgr B2, Sgr C, the 20 km s$^{-1}$ and 50 km s$^{-1}$ molecular clouds, and the Brick) in the Galactic central molecular zone (CMZ), and compare their star-forming activities with those in molecular clouds outside the CMZ. Using multi-band continuum observations taken from ${\it Planck}$, ${\it Herschel}$, JCMT/SCUBA-2, and CSO/SHARC2, we derived 8.5" resolution column density maps for the CMZ clouds and evaluated the column density probability distribution functions (N-PDFs). With the archival Atacama Large Millimeter/submillimeter Array (ALMA) 1.3 mm dust continuum data, we further evaluated the mass of the most massive cores ($M_{\rm core}^{\rm ma x}$). We find that the N-PDFs of four of the selected CMZ clouds are well described by a piecewise log-normal + power-law function, while the N-PDFs of the remaining two can be approximated by log-normal functions. In the first four targets, the masses in the power-law component ($M_{\rm gas}^{\rm bound}$), $M_{\rm core}^{\rm max}$, and star formation rate (SFR) are correlated. These correlations are very similar to those derived from low-mass clouds in the Solar neighborhood and massive star-forming regions on the Galactic disk. These findings lead to our key hypotheses: (1) In the extreme environment of the CMZ, the power-law component in the N-PDF also represents self-gravitationally bound gas structures, and (2) evolution and star-forming activities of self-gravitationally bound gas structures may be self-regulated, insensitive to the exterior environment on $\gtrsim$5-10 pc scales.

astro-ph.GA↗

arXiv2025

Deep Andromeda JCMT-SCUBA2 Observations. The Submillimeter Maps and Giant Molecular Clouds

We have carried out unprecedentedly deep, nearly confusion-limited JCMT-SCUBA2 mapping observations on the nearest spiral galaxy, M31 (Andromeda). The 850 $\mu$m image with a $\sim$50 pc resolution yields a comprehensive catalog of 383 giant molecular clouds (GMCs) that are associated with the spiral arms. In addition, it unveiled a population of 189 compact inter-arm GMCs in M31, which are mostly unresolved or marginally resolved. The masses of all these GMCs are in the range of 2$\times$10$^4$ -- 6$\times$10$^6$ $M_{\odot}$; the sizes are in the range of 30--130 pc. They follow a mass-size correlation, $M$ $\propto$ $R_{c}$$^{2.5}$. The inter-arm GMCs are systematically less massive, more diffuse, colder, and have lower star-forming efficiency (SFE) than on-arm GMCs. Moreover, within individual spatially resolved on-arm and off-arm M31 GMCs, the SFE is considerably lower than the SFE in molecular clouds in main sequence and green valley galaxies. Follow-up investigations on M31 GMCs may provide clues for how star formation may be quenched in galactic environments. Finally, we reconstrained the dust opacity spectral index $\beta$ in the M31 galaxy by combining our new JCMT observations with archival Herschel and Planck data and found that the radial variation of $\beta$ may not be as large as was proposed by previous studies.

astro-ph.GA↗

arXiv2025

Nascent chemical complexity in prestellar core IRAS 16293E: complex organics and deuterated methanol

Prestellar cores represent early sites of low-mass ($M$ $\leq$ few M$_\odot$) star and planet formation and provide insight into initial chemical conditions of complex organic molecules (COMs). Deuterated COMs trace the degree of molecular inheritance and/or reprocessing, as high deuteration in protostellar systems suggests COMs forming during the prestellar stage when deuteration is enhanced. Within the L1689N molecular cloud, the prestellar core IRAS 16293E sits $90^{"}$ eastward of the chemically-rich IRAS 16293-2422 A and B protostellar system. A unique view of star formation inside a common natal cloud, IRAS 16293A, B, and E all show some of the highest levels of deuteration in the ISM, with a number of D/H ratios $10^{5}$ times higher than Solar. We investigate for the first time the deuteration levels of the simplest COM, methanol (CH$_3$OH), in IRAS 16293E. Using the Arizona Radio Observatory (ARO) 12 m telescope, we target favorable transitions of CH$_2$DOH, CHD$_2$OH, $^{13}$CH$_3$OH, and several higher complexity COMs (including acetaldehyde, CH$_3$CHO, methyl formate, HCOOCH$_3$, and dimethyl ether, CH$_3$OCH$_3$) in the 3 mm band. Follow-up observations with the Yebes 40 m telescope provided additional transitions in the 7 mm (Q-band). We report the first detections of these COMs and deuterated methanol in prestellar core IRAS 16293E and use our observations to calculate excitation temperatures, column densities, and relative abundance ratios. Striking similarities are found between relative molecular ratios and D/H values when comparing IRAS 16293E to the A and B protostars, as well as to a heterogeneous sample of other prestellar cores, protostars, and the comet 67P/Churyumov-Gerasimenko. Our results support the idea that there is a limited amount of chemical reprocessing of COMs when prestellar cores collapse and heat-up during the protostellar phase.

astro-ph.GA↗

arXiv2025

HairShifter: Consistent and High-Fidelity Video Hair Transfer via Anchor-Guided Animation

Hair transfer is increasingly valuable across domains such as social media, gaming, advertising, and entertainment. While significant progress has been made in single-image hair transfer, video-based hair transfer remains challenging due to the need for temporal consistency, spatial fidelity, and dynamic adaptability. In this work, we propose HairShifter, a novel "Anchor Frame + Animation" framework that unifies high-quality image hair transfer with smooth and coherent video animation. At its core, HairShifter integrates a Image Hair Transfer (IHT) module for precise per-frame transformation and a Multi-Scale Gated SPADE Decoder to ensure seamless spatial blending and temporal coherence. Our method maintains hairstyle fidelity across frames while preserving non-hair regions. Extensive experiments demonstrate that HairShifter achieves state-of-the-art performance in video hairstyle transfer, combining superior visual quality, temporal consistency, and scalability. The code will be publicly available. We believe this work will open new avenues for video-based hairstyle transfer and establish a robust baseline in this field.

cs.CV↗

arXiv2025

Largest zero-dimensional intersection of $r$ degree $d$ hypersurfaces

Suppose we have $r$ hypersurfaces in $\mathbb{P}^m$ of degree $d$, whose defining polynomials are linearly independent, and their intersection has dimension $0$. Then what is the largest possible intersection of the $r$ hypersurfaces? We conjecture an exact formula for this problem and prove it when $m=2$. We show that this can be used to compute the generalized hamming weights of the projective Reed-Muller code $\operatorname{PRM}_q(d,2)$ and hence settle a conjecture of Beelen, Datta and Ghorpade for $m=2$.

math.AG↗