On the Computational Complexity of Polynomials and Bilinear Mappings. A Survey

This a r t i c l e is d e d i c a t e d to t h e s t u d y of t h e complex i ty of c o m p u t i n g a n d dec id ing s e l ec t ed p r o b l e m s in a lgebra , The m e t h o d s used he re a re a lgeb ra i ca l a n d (a lgebra ica l ly ) g e o m e t r i c a l in n a t u r e . The p r o b l e m s d e a l t wi th may be looked a t as m a t h e m a t i c a l mode l s of q u e s t i o n s ar ising in t h e o r e t i c a l c o m p u t e r sc ience . This, however, does n o t c o r r e s p o n d exac t ly to t h e h i s to r i cal d e v e l o p m e n t of a l g e b r a i c complex i ty theory , where a s p e c t s of n u m e r i c a l ana lys i s p lay a role too [ S t r a s s e n 1984]. As fa r as p r a c t i c a l a p p l i c a t i o n s a re c o n c e r n e d , i t is c e r t a i n l y t h e d e v e l o p m e n t of fas t a n d eff icient a l g o r i t h m s which is p r o m i n e n t . Here t h e m a t h e m a t i c i a n is i n t e r e s t e d m a i n l y in s u c h a l g o r i t h m s which a r e b a s e d on m a t h e m a t i c a l p rob lems . But t h i s a lone does n o t c o n s t i t u t e c o m p l e x i t y t h e o r y in i t s m o d e r n sense. I t is d e d i c a t e d to t h e s y s t e m a t i c s t u d y of t h e c o m p l e x i t y of p r o b l e m s known to be so lvable by a lgor i thms . Firs t , t h i s r e q u i r e s t h e d e v e l o p m e n t of m a t h e m a t i c a l l y p rec i se mode l s of complex i ty which o f t en a re s impl i f i ca t ions of t h e u s e r ' s c o n c e p t i o n s of eff iciency an d i t i n c l u d e s t h e q u e s t i o n w h e t h e r a g iven p r o c e d u r e is op t imal .

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