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Let $(\\ )^*:\\Bbb F_q({\\tt x}_0,\\dots,{\\tt x}_{n-1})\\to\\Bbb F_q({\\tt x}_0,\\dots,{\\tt x}_{n-1})$ be the $\\Bbb F_q$-monomorphism defined by ${\\tt x}_i^*={\\tt x}_{i+1}$ for $0\\le i< n-1$ and ${\\tt x}_{n-1}^*={\\tt x}_0^q$. For $f,g\\in\\Bbb F_q({\\tt x}_0,\\dots,{\\tt x}_{n-1})\\setminus\\Bbb F_q$, define $f\\circ g=f(g,g^*,\\dots,g^{(n-1)*})$. 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