We present an imaging survey of the CO(1--0), HCN(1--0), and HCO$^+$(1--0) lines in the nearby Seyfert galaxies using the Nobeyama Millimeter Array and RAINBOW Interferometer. Some of the observed Seyfert galaxies including NGC 1068, NGC 1097, NGC 5033, and NGC 5194 exhibit strong HCN(1--0) emission on a few 100 pc scales. The observed HCN(1--0)/CO(1--0) and HCN(1--0)/HCO$^+$(1--0) line ratios in the Seyfert nuclei ($>$0.2 and $>$1.8, respectively) have never been observed in the central regions of nuclear starburst galaxies. On the other hand, the molecular line ratios in the nuclei of NGC 3079, NGC 3227, NGC 4051, NGC 6764, NGC 7479, and NGC 7469 are comparable with those in the nuclear starburst galaxies. We propose that the elevated HCN emission originates from the X-ray irradiated dense molecular tori or XDRs close to the active nuclei. Our HCN/CO and HCN/HCO$^+$ diagrams will provide a new powerful diagnostic of the nuclear power source in active galaxies. Based on our diagnostic, we observe 3 of 5 type-1 Seyferts (6 of 10 in total) host compact nuclear starbursts. Our results are also supported by observations at other wavelengths such as those by L-band PAH spectroscopy. The proposed method will be crucial for investigating extremely dusty nuclei, such as ULIRGs and high-z submm galaxies, because these molecular lines are devoid of dust extinction. As an example, we present the HCN and HCO$^+$ observations of the LIRG NGC 4418, which suggests the presence of a buried active nucleus.
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