Polynomial maps on vector spaces over a finite field

Let l be a finite field of cardinality q and let n   be in Z≥1Z≥1. Let f1,…,fn∈l[x1,…,xn]f1,…,fn∈l[x1,…,xn] not all constant and consider the evaluation map f=(f1,…,fn):ln→lnf=(f1,…,fn):ln→ln. Set deg⁡(f)=maxi⁡deg⁡(fi)deg⁡(f)=maxi⁡deg⁡(fi). Assume that ln∖f(ln)ln∖f(ln) is not empty. We will prove|ln∖f(ln)|≥n(q−1)deg⁡(f). This improves previous known bounds.