no. G4
Géométrie algébrique
The set of forms with bounded strength is not closed
Ballico, Edoardo1 ; Bik, Arthur2  ; Oneto, Alessandro1 ; Ventura, Emanuele3
1 Università di Trento, Via Sommarive, 14 - 38123 Povo (Trento), Italy
2 MPI for Mathematics in the Sciences, Leipzig, Germany
3 Politecnico di Torino, Dipartimento di Scienze Matematiche “G. L. Lagrange”, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Comptes Rendus. Mathématique, Tome 360 (2022) no. G4, pp. 371-380.

The strength of a homogeneous polynomial (or form) is the smallest length of an additive decomposition expressing it whose summands are reducible forms. Using polynomial functors, we show that the set of forms with bounded strength is not always Zariski-closed. More specifically, if the ground field is algebraically closed, we prove that the set of quartics with strength ≤3 is not Zariski-closed for a large number of variables.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.302
Classification : 15A21, 13A02, 14R20
Ballico, Edoardo 1 ; Bik, Arthur 2 ; Oneto, Alessandro 1 ; Ventura, Emanuele 3

1 Università di Trento, Via Sommarive, 14 - 38123 Povo (Trento), Italy
2 MPI for Mathematics in the Sciences, Leipzig, Germany
3 Politecnico di Torino, Dipartimento di Scienze Matematiche “G. L. Lagrange”, Corso Duca degli Abruzzi 24, 10129 Torino, Italy 
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     title = {The set of forms with bounded strength is not closed},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {371--380},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
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     year = {2022},
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Ballico, Edoardo; Bik, Arthur; Oneto, Alessandro; Ventura, Emanuele. The set of forms with bounded strength is not closed. Comptes Rendus. Mathématique, Tome 360 (2022) no. G4, pp. 371-380. doi : 10.5802/crmath.302. http://www.numdam.org/articles/10.5802/crmath.302/

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