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We consider a completely resonant nonlinear Schr\"odinger equation on the$d$-dimensional torus, for any $d\geq 1$, with polynomial nonlinearity of anydegree $2p+1$, $p\geq1$, which is gauge and translation invariant. We study thebehaviour of high Sobolev $H^{s}$-norms of solutions, $s\geq s_1+1 > d/2 + 2$,whose initial datum $u_0\in H^{s}$ satisfies an appropriate smallness conditionon its low $H^{s_1}$ and $L^2$-norms respectively. We prove a polynomial upperbound on the possible growth of the Sobolev norm $H^{s}$ over finite but longtime scale that is exponential in the regularity parameter $s_1$. As abyproduct we get stability of the low $H^{s_1}$-norm over such time interval. Akey ingredient in the proof is the introduction of a suitable ``modifiedenergy" that provides an a priori upper bound on the growth. This is obtainedby combining para-differential techniques and suitable tame estimates.

On the lifespan of solutions and control of high Sobolev norms for the completely resonant NLS on tori