-- Copyright (c) David Amos, 2011. All rights reserved. {-# LANGUAGE TupleSections, NoMonomorphismRestriction #-} -- |A module providing an efficient implementation of the Buchberger algorithm for calculating the (reduced) Groebner basis for an ideal, -- together with some straightforward applications. module Math.CommutativeAlgebra.GroebnerBasis where import Data.List as L import qualified Data.IntMap as IM import qualified Data.Set as S import Math.Core.Utils import Math.Core.Field import Math.Algebras.VectorSpace import Math.Algebras.Structures import Math.CommutativeAlgebra.Polynomial -- Sources: -- Cox, Little, O'Shea: Ideals, Varieties and Algorithms -- Giovini, Mora, Niesi, Robbiano, Traverso, "One sugar cube please, or Selection strategies in the Buchberger algorithm" sPoly f g = let d = tgcd (lt f) (lt g) in (lt g `tdiv` d) *-> f - (lt f `tdiv` d) *-> g -- The point about the s-poly is that it cancels out the leading terms of the two polys, exposing their second terms isGB fs = all (\h -> h %% fs == 0) (pairWith sPoly fs) -- Initial, naive version -- Cox p87 gb1 fs = gb' fs (pairWith sPoly fs) where gb' gs (h:hs) = let h' = h %% gs in if h' == 0 then gb' gs hs else gb' (h':gs) (hs ++ map (sPoly h') gs) gb' gs [] = gs -- [f xi xj | xi <- xs, xj <- xs, i < j] pairWith f (x:xs) = map (f x) xs ++ pairWith f xs pairWith _ [] = [] -- Cox p89-90 reduce gs = reduce' gs [] where reduce' (l:ls) rs = let l' = l %% (rs ++ ls) in if l' == 0 then reduce' ls rs else reduce' ls (toMonic l':rs) reduce' [] rs = L.sort rs -- when using an elimination order, the elimination ideal will be at the end -- Cox et al p106-7 -- No need to calculate an spoly fi fj if -- 1. the lm fi and lm fj are coprime, or -- 2. there exists some fk, with (i,k) (j,k) already considered, and lm fk divides lcm (lm fi) (lm fj) -- some slight inefficiencies from looking up fi, fj repeatedly gb2 fs = reduce $ gb' fs (pairs [1..s]) s where s = length fs gb' gs ((i,j):ps) t = let fi = gs!i; fj = gs!j in if mcoprime (lm fi) (lm fj) || criterion fi fj then gb' gs ps t else let h = sPoly fi fj %% gs in if h == 0 then gb' gs ps t else gb' (gs++[h]) (ps ++ [ (i,t+1) | i <- [1..t] ]) (t+1) where criterion fi fj = let l = mlcm (lm fi) (lm fj) in any (test l) [1..t] test l k = k `notElem` [i,j] && ordpair i k `notElem` ps && ordpair j k `notElem` ps && lm (gs!k) `mdivides` l gb' gs [] _ = gs xs ! i = xs !! (i-1) -- in other words, index the list from 1 not 0 -- Doesn't result in any speedup gb2a fs = reduce $ gb' fs' (pairs [1..s]) s where fs' = IM.fromList $ zip [1..] $ filter (/= 0) fs s = IM.size fs' gb' gs ((i,j):ps) t = let fi = gs IM.! i; fj = gs IM.! j in if mcoprime (lm fi) (lm fj) || criterion fi fj then gb' gs ps t else let h = sPoly fi fj %% IM.elems gs in if h == 0 then gb' gs ps t else let t' = t+1 gs' = IM.insert t' h gs ps' = ps ++ map (,t') [1..t] in gb' gs' ps' t' where criterion fi fj = let l = mlcm (lm fi) (lm fj) in any (test l) [1..t] test l k = k `notElem` [i,j] && ordpair i k `notElem` ps && ordpair j k `notElem` ps && lm (gs IM.! k) `mdivides` l gb' gs [] _ = IM.elems gs -- version of gb2 where we eliminate pairs as they're created, rather than as they're processed gb3 fs = reduce $ gb1' [] fs [] 0 where gb1' gs (f:fs) ps t = let ps' = updatePairs gs ps f t in gb1' (gs ++ [f]) fs ps' (t+1) gb1' ls [] ps t = gb2' ls ps t gb2' gs ((i,j):ps) t = let h = sPoly (gs!i) (gs!j) %% gs in if h == 0 then gb2' gs ps t else let ps' = updatePairs gs ((i,j):ps) h t in gb2' (gs++[h]) ps' (t+1) gb2' gs [] _ = gs updatePairs gs ps f t = [p | p@(i,j) <- ps, not (lm f `mdivides` mlcm (lm (gs!i)) (lm (gs!j)))] ++ [ (i,t+1) | (gi,i) <- zip gs [1..t], not (mcoprime (lm gi) (lm f)), not (criterion (mlcm (lm gi) (lm f)) i) ] where criterion l i = any (`mdivides` l) [lm gk | (gk,k) <- zip gs [1..t], k /= i, ordpair i k `notElem` ps] -- Cox et al 108 -- 1. list smallest fs first, as more likely to reduce -- 2. order the pairs with smallest lcm fi fj first ("normal selection strategy") gb4 fs = reduce $ gb1' [] fs' [] 0 where fs' = reverse $ L.sort $ filter (/=0) fs gb1' gs (f:fs) ps t = gb1' (gs ++ [f]) fs ps' (t+1) where ps' = updatePairs gs ps f t gb1' ls [] ps t = gb2' ls ps t gb2' gs ((l,(i,j)):ps) t = let h = sPoly (gs!i) (gs!j) %% gs in if h == 0 then gb2' gs ps t else let ps' = updatePairs gs ((l,(i,j)):ps) h t in gb2' (gs++[h]) ps' (t+1) gb2' gs [] _ = gs updatePairs gs ps f t = let oldps = [p | p@(l,(i,j)) <- ps, not (lm f `mdivides` l)] newps = sortBy (flip cmpfst) [ (l,(i,t+1)) | (gi,i) <- zip gs [1..t], let l = mlcm (lm gi) (lm f), not (mcoprime (lm gi) (lm f)), not (criterion l i) ] in mergeBy (flip cmpfst) oldps newps where criterion l i = any (`mdivides` l) [lm gk | (gk,k) <- zip gs [1..t], k /= i, ordpair i k `notElem` map snd ps] mergeBy cmp (t:ts) (u:us) = case cmp t u of LT -> t : mergeBy cmp ts (u:us) EQ -> t : mergeBy cmp ts (u:us) GT -> u : mergeBy cmp (t:ts) us mergeBy _ ts us = ts ++ us -- one of them is null -- Giovini et al -- The point of sugar is, given fi, fj, to give an upper bound on the degree of sPoly fi fj without having to calculate it -- We can then select by preference pairs with lower sugar, expecting therefore that the s-polys will have lower degree -- It is only for Lex ordering that sugar seems to give a substantial improvement -- gb4 is fine for Grevlex -- !! can probably get rid of the Ord k requirement in the following -- |Given a list of polynomials over a field, return a Groebner basis for the ideal generated by the polynomials. gb :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] gb fs = let fs' = reverse $ sort $ map toMonic $ filter (/=0) fs in reduce $ gb1' [] fs' [] 0 where gb1' gs (f:fs) ps t = gb1' (gs ++ [f]) fs ps' (t+1) where ps' = updatePairs gs ps f (t+1) gb1' ls [] ps t = gb2' ls ps t gb2' gs (p@(_,(i,j)):ps) t = if h == 0 then gb2' gs ps t else gb2' (gs++[h]) ps' (t+1) where h = toMonic $ sPoly (gs!i) (gs!j) %% gs ps' = updatePairs gs (p:ps) h (t+1) gb2' gs [] _ = gs updatePairs gs ps gk k = let newps = [let l = mlcm (lm gi) (lm gk) in ((sugar gi gk l, l), (i,k)) | (gi,i) <- zip gs [1..k-1] ] ps' = [p | p@((sij,tij),(i,j)) <- ps, let ((sik,tik),_) = newps ! i, let ((sjk,tjk),_) = newps ! j, not ( (tik `mproperlydivides` tij) && (tjk `mproperlydivides` tij) ) ] -- sloppy variant newps' = discard1 [] newps newps'' = sortBy (flip cmpSug) $ discard2 [] $ sortBy (flip cmpNormal) newps' in mergeBy (flip cmpSug) ps' newps'' where discard1 ls (r@((_sik,tik),(i,_k)):rs) = if lm (gs!i) `mcoprime` lm gk -- then discard [l | l@((_,tjk),_) <- ls, tjk /= tik] [r | r@((_,tjk),_) <- ls, tjk /= tik] then discard1 (filter (\((_,tjk),_) -> tjk /= tik) ls) (filter (\((_,tjk),_) -> tjk /= tik) rs) else discard1 (r:ls) rs discard1 ls [] = ls discard2 ls (r@((_sik,tik),(i,k)):rs) = discard2 (r:ls) $ filter (\((_sjk,tjk),(j,k')) -> not (k == k' && tik `mdivides` tjk)) rs discard2 ls [] = ls -- The two calls to toMonic are designed to prevent coefficient explosion, but it is unproven that they are effective -- sugar of sPoly f g, where h = lcm (lt f) (lt g) -- this is an upper bound on deg (sPoly f g) sugar f g h = mdeg h + max (deg f - mdeg (lm f)) (deg g - mdeg (lm g)) cmpNormal ((s1,t1),(i1,j1)) ((s2,t2),(i2,j2)) = compare (t1,j1) (t2,j2) cmpSug ((s1,t1),(i1,j1)) ((s2,t2),(i2,j2)) = compare (-s1,t1,j1) (-s2,t2,j2) {- merge (t:ts) (u:us) = if t <= u then t : merge ts (u:us) else u : merge (t:ts) us merge ts us = ts ++ us -- one of them is null -} -- OPERATIONS ON IDEALS memberGB f gs = f %% gs == 0 -- |@memberI f gs@ returns whether f is in the ideal generated by gs memberI :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => Vect k m -> [Vect k m] -> Bool memberI f gs = memberGB f (gb gs) -- Cox et al, p181 -- |Given ideals I and J, their sum is defined as I+J = {f+g | f \<- I, g \<- J}. -- -- If fs and gs are generators for I and J, then @sumI fs gs@ returns generators for I+J. -- -- The geometric interpretation is that the variety of the sum is the intersection of the varieties, -- ie V(I+J) = V(I) intersect V(J) sumI :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> [Vect k m] sumI fs gs = gb (fs ++ gs) -- Cox et al, p183 -- |Given ideals I and J, their product I.J is the ideal generated by all products {f.g | f \<- I, g \<- J}. -- -- If fs and gs are generators for I and J, then @productI fs gs@ returns generators for I.J. -- -- The geometric interpretation is that the variety of the product is the union of the varieties, -- ie V(I.J) = V(I) union V(J) productI :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> [Vect k m] productI fs gs = gb [f * g | f <- fs, g <- gs] -- Cox et al, p185-6 -- |The intersection of ideals I and J is the set of all polynomials which belong to both I and J. -- -- If fs and gs are generators for I and J, then @intersectI fs gs@ returns generators for the intersection of I and J -- -- The geometric interpretation is that the variety of the intersection is the union of the varieties, -- ie V(I intersect J) = V(I) union V(J). -- -- The reason for prefering the intersection over the product is that the intersection of radical ideals is radical, -- whereas the product need not be. intersectI :: (Fractional k, Ord k, Monomial m, Ord m) => [Vect k m] -> [Vect k m] -> [Vect k m] intersectI fs gs = let t = toElimFst $ return $ (mvar "t" :: Glex String) hs = map ((t *) . toElimSnd) fs ++ map (((1-t) *) . toElimSnd) gs in eliminateFst hs toElimFst = fmap (\m -> Elim2 m munit) toElimSnd = fmap (\m -> Elim2 munit m) isElimFst = (/= munit) . (\(Elim2 m _) -> m) . lm fromElimSnd = fmap (\(Elim2 _ m) -> m) eliminateFst = map fromElimSnd . dropWhile isElimFst . gb -- Cox et al, p193-4 -- |Given ideals I and J, their quotient is defined as I:J = {f | f \<- R, f.g is in I for all g in J}. -- -- If fs and gs are generators for I and J, then @quotientI fs gs@ returns generators for I:J. -- -- The ideal quotient is the algebraic analogue of the Zariski closure of a difference of varieties. -- V(I:J) contains the Zariski closure of V(I)-V(J), with equality if k is algebraically closed and I is a radical ideal. quotientI :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> [Vect k m] quotientI _ [] = [1] quotientI fs gs = foldl1 intersectI $ map (quotientP fs) gs -- quotientI fs gs = foldl intersectI [1] $ map (quotientP fs) gs quotientP fs g = map ( // g ) $ intersectI fs [g] where h // g = let ([u],_) = quotRemMP h [g] in u -- |@eliminate vs gs@ returns the elimination ideal obtained from the ideal generated by gs by eliminating the variables vs. eliminate :: (Eq k, Fractional k, Ord k, MonomialConstructor m, Monomial (m v), Ord (m v)) => [Vect k (m v)] -> [Vect k (m v)] -> [Vect k (m v)] eliminate vs gs = let subs = subFst vs in eliminateFst [g `bind` subs | g <- gs] where subFst :: (Eq k, Num k, MonomialConstructor m, Eq (m v), Mon (m v)) => [Vect k (m v)] -> v -> Vect k (Elim2 (m v) (m v)) subFst vs = (\v -> let v' = var v in if v' `elem` vs then toElimFst v' else toElimSnd v') {- -- !! NOT WORKING -- |@elimExcept vs gs@ returns the elimination ideal obtained from the ideal generated by gs by eliminating all variables except vs. elimExcept :: (Fractional k, Ord k, MonomialConstructor m, Monomial (m v), Ord (m v)) => [Vect k (m v)] -> [Vect k (m v)] -> [Vect k (m v)] elimExcept vs gs = let subs = subSnd vs in eliminateFst [g `bind` subs | g <- gs] where subSnd :: (Num k, MonomialConstructor m, Eq (m v), Mon (m v)) => [Vect k (m v)] -> v -> Vect k (Elim2 (m v) (m v)) subSnd vs = (\v -> let v' = var v in if v' `elem` vs then toElimSnd v' else toElimFst v') -} -- MONOMIAL BASES FOR QUOTIENT ALGEBRAS -- basis for the polynomial ring in variables vs mbasis vs = mbasis' [1] where mbasis' ms = ms ++ mbasis' (toSet [v*m | v <- vs, m <- ms]) -- |Given variables vs, and a Groebner basis gs, @mbasisQA vs gs@ returns a monomial basis for the quotient algebra k[vs]/\<gs\>. -- For example, @mbasisQA [x,y] [x^2+y^2-1]@ returns a monomial basis for k[x,y]/\<x^2+y^2-1\>. -- In general, the monomial basis is likely to be infinite. mbasisQA :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> [Vect k m] mbasisQA vs gs = mbasisQA' [1] where mbasisQA' [] = [] -- the quotient algebra is finite-dimensional mbasisQA' ms = ms ++ mbasisQA' (toSet [f | v <- vs, m <- ms, let f = v*m, f %% gs == f]) -- |Given an ideal I, the leading term ideal lt(I) consists of the leading terms of all elements of I. -- If I is generated by gs, then @ltIdeal gs@ returns generators for lt(I). ltIdeal :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] ltIdeal gs = map (return . lm) $ gb gs -- number of monomials of degree i in n variables numMonomials n i = toInteger (i+n-1) `choose` toInteger (n-1) -- |Given variables vs, and a homogeneous ideal gs, @hilbertFunQA vs gs@ returns the Hilbert function for the quotient algebra k[vs]/\<gs\>. -- Given an integer i, the Hilbert function returns the number of degree i monomials in a basis for k[vs]/\<gs\>. -- For a homogeneous ideal, this number is independent of the monomial ordering used -- (even though the elements of the monomial basis themselves are dependent on the ordering). -- -- If the ideal I is not homogeneous, then R/I is not graded, and the Hilbert function is not well-defined. -- Specifically, the number of degree i monomials in a basis is likely to depend on which monomial ordering you use. hilbertFunQA :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> Int -> Integer hilbertFunQA vs gs i = hilbertFunQA' (ltIdeal gs) i where n = length vs hilbertFunQA' _ i | i < 0 = 0 hilbertFunQA' (m:ms) i = hilbertFunQA' ms i - hilbertFunQA' (ms `quotientP` m) (i - deg m) hilbertFunQA' [] i = numMonomials n i -- For example, consider k[x,y]/<x-y^2> -- Under Lex ordering, the monomial basis is 1,y,y^2,y^3,... -- Under Glex ordering, the monomial basis is 1,x,y,x^2,xy,x^3,x^2y,... -- So the Hilbert function is not well-defined. -- Note though that this function does still correctly return the number of degree i monomials for the given monomial ordering -- naive implementation which simply counts monomials hilbertSeriesQA1 vs gs = hilbertSeriesQA1' [1] where hilbertSeriesQA1' [] = [] -- repeat 0 hilbertSeriesQA1' ms = length ms : hilbertSeriesQA1' (toSet [f | v <- vs, m <- ms, let f = v*m, f %% gs == f]) -- Eisenbud p325, p357 / Schenck p56 -- This can be made more efficient by choosing which m to recurse on -- |Given variables vs, and a homogeneous ideal gs, @hilbertSeriesQA vs gs@ returns the Hilbert series for the quotient algebra k[vs]/\<gs\>. -- The Hilbert series should be interpreted as a formal power series where the coefficient of t^i is the Hilbert function evaluated at i. -- That is, the i'th element in the series is the number of degree i monomials in a basis for k[vs]/\<gs\>. hilbertSeriesQA :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> [Integer] hilbertSeriesQA vs gs = hilbertSeriesQA' $ ltIdeal gs where hilbertSeriesQA' (m:ms) = hilbertSeriesQA' ms <-> (replicate (deg m) 0 ++ hilbertSeriesQA' (ms `quotientI` [m])) hilbertSeriesQA' [] = [numMonomials n i | i <- [0..] ] n = length vs (a:as) <-> (b:bs) = (a-b) : (as <-> bs) as <-> [] = as [] <-> bs = map negate bs -- |In the case where every variable v occurs in some generator g of the homogeneous ideal (the usual case), -- then the vs can be inferred from the gs. -- @hilbertSeriesQA' gs@ returns the Hilbert series for the quotient algebra k[vs]/\<gs\>. hilbertSeriesQA' :: (Fractional k, Ord k, MonomialConstructor m, Ord (m v), Monomial (m v), Algebra k (m v)) => [Vect k (m v)] -> [Integer] hilbertSeriesQA' gs = hilbertSeriesQA vs gs where vs = toSet (concatMap vars gs) -- |For i \>\> 0, the Hilbert function becomes a polynomial in i, called the Hilbert polynomial. hilbertPolyQA :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> GlexPoly Q String hilbertPolyQA vs gs = hilbertPolyQA' (ltIdeal gs) i where n = toInteger $ length vs i = glexvar "i" hilbertPolyQA' [] x = product [ x + fromInteger j | j <- [1..n-1] ] / (fromInteger $ product [1..n-1]) hilbertPolyQA' (m:ms) x = hilbertPolyQA' ms x - hilbertPolyQA' (ms `quotientP` m) (x - fromIntegral (deg m)) hilbertPolyQA' :: (Fractional k, Ord k, MonomialConstructor m, Ord (m v), Monomial (m v), Algebra k (m v)) => [Vect k (m v)] -> GlexPoly Q String hilbertPolyQA' gs = hilbertPolyQA vs gs where vs = toSet (concatMap vars gs) -- The dimension of a variety dim vs gs = 1 + deg (hilbertPolyQA vs gs) dim' gs = 1 + deg (hilbertPolyQA' gs)