We imitate a classical construction of a counterexample to the local-global principle of cubic forms of 4 variables which was discovered first by Swinnerton-Dyer (Mathematika (1962)). Our construction gives new explicit families of counterexamples in homogeneous forms of variables of degree for infinitely many integers . It is contrastive to Swinnerton-Dyer’s original construction that we do not need any concrete calculation in the proof of local solubility.
"Counterexamples to the local-global principle associated with Swinnerton-Dyer's cubic form." Rocky Mountain J. Math. 50 (6) 2097 - 2102, December 2020. https://doi.org/10.1216/rmj.2020.50.2097