A system of n algebraic equations for n unknowns is considered, in which the collection of exponents is fixed, and the coefficients are variable. Since the solutions of such systems are 2n-homogeneous, two coefficients in each equation can be fixed, which makes it possible to pass to the corresponding reduced systems. For the reduced systems, a formula for the solution (and also for any monomial of the solution) is obtained in the form of a hypergeometric type series in the coefficients. Such series are represented as a finite sum of Horn's hypergeometric series: the ratios of the neighboring coefficients of the latter series are rational functions of summation variables. The study is based on the linearization procedure and on the theory of multidimensional residues. As an application of the main formula, a multidimensional analog is presented of the Waring formula for powers of the roots of the system.

The spatial arrangement of interfaces between homeodomain transcription factors and operator DNA has been considered. We analyzed the binding contacts for a representative set of 22 complexes of homeodomain transcription factors with a double-stranded operator DNA in the region of the major groove. It was shown that the recognition of DNA by the recognizing _-helix of protein is governed by two contact groups. Invariant protein-DNA group of contacts includes six contacts, formed by atomic groups of coding and non-coding DNA chains with the groups of amino acids. The recognizing _-helix forms contacts by polar groups of residues Trp2 (NE1), Asn5, and Lys9 with the canonical sequence T(1)A(2)A(3)T(4) of the coding DNA chain, and contacts by residues Lys0, Arg7 and Lys11 with the sequence A(4)X(5)X(6)X(7) of a non-coding DNA chain, where X is any nucleotide. Variable protein-DNA group of contacts comprises two groups bound with the sequence T(3)A(4)X(5)X(6) of the non- coding DNA-chain. These contacts are mainly with the bases and specify the binding pattern of individual homeodomains. The invariant contact group represents a recognition pattern for transcription factors of the homeodomain family: multiple adenine-asparagine contact and six position-specific phosphate contacts mainly with lysine or arginine. Within this group, we have found three most significant invariant contacts which allow deducing the recognition rules for homeodomains. These rules are inherent for different taxonomic groups of the homeodomain family and can distinguishing members of this family from any other family of transcription factors.

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