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The tensorial equations for non trivial fully interacting fixed points atlowest order in the $\varepsilon$ expansion in $4-\varepsilon$ and$3-\varepsilon$ dimensions are analysed for $N$-component fields andcorresponding multi-index couplings $\lambda$ which are symmetric tensors withfour or six indices. Both analytic and numerical methods are used. For$N=5,6,7$ in the four-index case large numbers of irrational fixed points arefound numerically where $||\lambda ||^2$ is close to the bound found by Rychkovand Stergiou in arXiv:1810.10541. No solutions, other than those already known,are found which saturate the bound. These examples in general do not haveunique quadratic invariants in the fields. For $N \geqslant 6$ the stabilitymatrix in the full space of couplings always has negative eigenvalues. In thesix index case the numerical search generates a very large number of solutionsfor $N=5$.

Heavy Handed Quest for Fixed Points in Multiple Coupling Scalar Theories in the $\varepsilon$ Expansion