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Detecting LLM-generated text in specialized and high-stakes domains like medicine and law is crucial for combating misinformation and ensuring authenticity. However, current zero-shot detectors, while effective on general text, often fail when applied to specialized content due to domain shift. We provide a theoretical analysis showing this failure is fundamentally linked to the KL divergence between human, detector, and source text distributions. To address this, we propose DivScore, a zero-shot detection framework using normalized entropy-based scoring and domain knowledge distillation to robustly identify LLM-generated text in specialized domains. Experiments on medical and legal datasets show that DivScore consistently outperforms state-of-the-art detectors, with 14.4% higher AUROC and 64.0% higher recall at 0.1% false positive rate threshold. In adversarial settings, DivScore demonstrates superior robustness to other baselines, achieving on average 22.8% advantage in AUROC and 29.5% in recall.