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We present new real algebraic maps of non-positive codimensions withprescribed images whose boundaries consist of explicit non-singular realalgebraic hypersurfaces intersecting with so-called "transversality". We alsohave the maps with explicit information on important real polynomials.  This gives new construction of explicit real algebraic maps of non-positivecodimensions with explicit information on images, preimages and important realpolynomials. Thanks to some celebrated theory of Nash, smooth closed manifoldsare {\it non-singular} real algebraic manifolds and the zero sets of some realpolynomials. In considerable cases, we can approximate smooth functions or moregenerally, maps, by real algebraic ones. It is in general difficult to haveexplicit examples. We have constructed some previously. We have ones of somenew type with explicit information on global structures of the maps.

Natural real algebraic maps of non-positive codimensions with prescribed  images whose boundaries consist of non-singular real algebraic hypersurfaces  intersecting with transversality