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On strong orthogonal systems and weak permutation polynomials over finite commutative rings. (English) Zbl 1109.11058
Let $$f_1, \dots, f_k$$ be $$k$$ polynomials in $$n$$ variables over a finite commutative ring $$R$$. If they induce a uniform map from $$R^n$$ to $$R^k$$ then they are said to form a weak orthogonal system over $$R$$ and they are said to form a strong orthogonal system over $$R$$ if there additionally exist polynomials $$f_{k+1},\dots, f_n$$ such that $$f_1,\dots,f_n$$ induce a permutation of $$R^n$$. If $$k = 1$$ then the polynomial in $$n$$ variables is called a weak (strong) permutation polynomial.
As main result the authors prove that $$k$$ polynomials $$f_1, \dots, f_k$$ in $$n$$ variables over a finite commutative local ring $$R$$ with maximal ideal $$M$$ generated by $$r$$ elements (where $$r$$ is chosen minimal) form a strong orthogonal system over $$R$$ if and only if $$f_1\bmod M, \dots, f_k\bmod M$$ form a weak orthogonal system over $$R/M$$ and the Jacobi matrix ($$f_1^\prime(x)\bmod M, \dots, f_k^\prime(x)\bmod M$$) has rank $$k$$ everywhere. Furthermore if $$n \leq r$$, then every weak permutation polynomial in $$R[X_1,\dots,X_n]$$ is strong.

MSC:
 11T06 Polynomials over finite fields 13M10 Polynomials and finite commutative rings 13B25 Polynomials over commutative rings
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References:
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