Zeros of special polynomials and their impact on a class of APN functions

In 2021, Calderini et al. introduced a construction for APN functions on ${\mathbb{F}}_{2^{2m}}$ in bivariate form $$f(x,y)=(xy,x^{2^r+1}+x^{2^{r+m\mathrm{/}2}}{y}^{2^{m\mathrm{/}2}}+bx{y}^{2^r}+c{y}^{2^r+1}),r<m\mathrm{/}2,\gcd (r,m)=1.$$ They showed that this family exists provided the existence of a polynomial $$P_{c,b}(X)=(c{X}^{2^r+1}+b{X}^{2^r}+1{)}^{2^{m\mathrm{/}2}+1}+X^{2^{m\mathrm{/}2}+1},$$ with no zeros in ${\mathbb{F}}_{2^{2m}}$. For $m\le 6$ it was shown that we can have APN functions belonging to this family. However, up to now, no construction of such polynomials is known for $m\ge 8$. In this work we provide a non-existence result of such functions whenever $r<m\mathrm{/}8-1$, by application of techniques from algebraic varieties over finite fields. In particular, for $r=1$ we have that the construction of Calderini et al. cannot provide an APN function for $m\ge 8$.