# The motivic Thom–Sebastiani theorem for regular and formal functions

• Quy Thuong Lê

## Abstract

Thanks to the work of Hrushovski and Loeser on motivic Milnor fibers, we give a model-theoretic proof for the motivic Thom–Sebastiani theorem in the case of regular functions. Moreover, slightly extending Hrushovski–Loeser’s construction adjusted to Sebag, Loeser and Nicaise’s motivic integration for formal schemes and rigid varieties, we formulate and prove an analogous result for formal functions. The latter is meaningful as it has been a crucial element of constructing Kontsevich–Soibelman’s theory of motivic Donaldson–Thomas invariants.

Award Identifier / Grant number: 246903/NMNAG

Funding statement: The author is partially supported by the Centre Henri Lebesgue (program “Investissements d’avenir”, ANR-11-LABX-0020-01) and by ERC under the European Community’s Seventh Framework Programme (FP7/2007-2013), ERC Grant Agreement no. 246903/NMNAG.

## Acknowledgements

The author is grateful to François Loeser, Julien Sebag and Michel Raibaut for useful discussions. He would like to thank the Centre Henri Lebesgue and the Université de Rennes 1 for awarding him a postdoctoral fellowship and an excellent atmosphere during his stay there.

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