source: branches/f4grobner/ideal.lisp@ 2612

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;;;  Copyright (C) 1999, 2002, 2009, 2015 Marek Rychlik <rychlik@u.arizona.edu>          
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;; Operations in ideal theory
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(defpackage "IDEAL"
  (:use :cl :ring :monom :order :term :polynomial :division :grobner-wrap :ring-and-order)
  (:export "POLY-DEPENDS-P"
           "RING-INTERSECTION" 
           "ELIMINATION-IDEAL"
           "COLON-IDEAL"
           "COLON-IDEAL-1"
           "IDEAL-INTERSECTION"
           "POLY-LCM"
           "GROBNER-EQUAL"
           "GROBNER-SUBSETP"
           "GROBNER-MEMBER"
           "IDEAL-SATURATION-1"
           "IDEAL-SATURATION"
           "IDEAL-POLYSATURATION-1"
           "IDEAL-POLYSATURATION"
           ))

(in-package :ideal)

;; Does the polynomial P depend on variable K?
(defun poly-depends-p (p k)
  "Return T if the term polynomial P depends on variable number K."
  (some #'(lambda (term) (term-depends-p term k)) (poly-termlist p)))

(defun ring-intersection (plist k)
  "This function assumes that polynomial list
PLIST=(P[0],P[1],...,P[J-1]) is a Grobner basis and it calculates the
intersection of Id({P[0],P[1],...,P[J-1]}) with the ring
R[X[K],...,X[N-1]], i.e.  it discards polynomials which depend on
variables X[0], X[1], ..., X[K-1]."
  (dotimes (i k plist)
    (setf plist
          (remove-if #'(lambda (p)
                         (poly-depends-p p i))
                     plist))))

(defun elimination-ideal (ring-and-order flist k 
                          &optional 
                            (top-reduction-only $poly_top_reduction_only)
                            (start 0))
  "Given a list of polynomials FLIST, and an integer K, tt finds and
returns the Groebner basis the elimination ideal of Id({FLIST})
obtained by eliminating the first K variables. Optional argument
TOP-REDUCTION-ONLY indicates whether to fully reduce or only
top-reduce. Optional argument START, defaulting to 0, is used to
indicate that the first START elements of F form a Groebner basis."
  (ring-intersection (reduced-grobner ring-and-order flist start top-reduction-only) k))

(defun colon-ideal (ring-and-order f g 
                    &optional 
                      (top-reduction-only $poly_top_reduction_only)
                    &aux
                      (ring (ro-ring ring-and-order)))
  "Returns the reduced Grobner basis of the colon ideal Id(F):Id(G),
where F and G are two lists of polynomials. The colon ideal I:J is
defined as the set of polynomials H such that for all polynomials W in
J the polynomial W*H belongs to I."
  (declare (type ring-and-order ring-and-order))
  (cond
    ((endp g)
     ;;Id(G) consists of 0 only so W*0=0 belongs to Id(F)
     (if (every #'poly-zerop f)
         (error "First ideal must be non-zero.")
         (list (make-poly-from-termlist
                (list (make-term
                       :monom (make-monom :dimension (monom-dimension (poly-lm (find-if-not #'poly-zerop f))))
                       :coeff (funcall (ring-unit ring))))))))
    ((endp (cdr g))
     (colon-ideal-1 ring-and-order f (car g) top-reduction-only))
    (t
     (ideal-intersection ring-and-order
                         (colon-ideal-1 ring-and-order f (car g) top-reduction-only)
                         (colon-ideal ring-and-order f (rest g) top-reduction-only)
                         top-reduction-only))))

(defun colon-ideal-1 (ring-and-order f g 
                      &optional 
                        (top-reduction-only $poly_top_reduction_only))
  "Returns the reduced Grobner basis of the colon ideal Id(F):Id({G}), where
F is a list of polynomials and G is a polynomial."
  (declare (type ring-and-order ring-and-order))
  (mapcar #'(lambda (x) 
              (poly-exact-divide ring-and-order x g)) 
          (ideal-intersection ring-and-order f (list g) top-reduction-only)))

(defun ideal-intersection (ring-and-order f g 
                           &optional 
                             (top-reduction-only $poly_top_reduction_only)
                             (ring (ro-ring ring-and-order)))
  (declare (type ring-and-order ring-and-order))
  (mapcar #'poly-contract
          (ring-intersection
           (reduced-grobner
            ring-and-order
            (append (mapcar #'(lambda (p) (poly-extend p (make-monom :dimension 1 :initial-exponent 1))) f)
                    (mapcar #'(lambda (p)
                                (poly-append (poly-extend (poly-uminus ring p)
                                                          (make-monom :dimension 1 :initial-exponent 1))
                                             (poly-extend p)))
                            g))
            0
            top-reduction-only)
           1)))

(defun poly-lcm (ring-and-order f g &aux (ring (ro-ring ring-and-order)))
  "Return LCM (least common multiple) of two polynomials F and G.
The polynomials must be ordered according to monomial order PRED
and their coefficients must be compatible with the RING structure
defined in the COEFFICIENT-RING package."
  (cond
    ((poly-zerop f) f)
    ((poly-zerop g) g)
    ((and (endp (cdr (poly-termlist f))) (endp (cdr (poly-termlist g))))
     (let ((m (monom-lcm (poly-lm f) (poly-lm g))))
       (make-poly-from-termlist (list (make-term :monom m 
                                                 :coeff (funcall (ring-lcm ring) (poly-lc f) (poly-lc g)))))))
    (t
     (multiple-value-bind (f f-cont)
         (poly-primitive-part ring f)
       (multiple-value-bind (g g-cont)
           (poly-primitive-part ring g)
         (scalar-times-poly
          ring
          (funcall (ring-lcm ring) f-cont g-cont)
          (poly-primitive-part ring (car (ideal-intersection ring-and-order (list f) (list g) nil)))))))))

;; Do two Grobner bases yield the same ideal?
(defun grobner-equal (ring-and-order g1 g2)
  "Returns T if two lists of polynomials G1 and G2, assumed to be Grobner bases,
generate  the same ideal, and NIL otherwise."
  (declare (type ring-and-order ring-and-order))
  (and (grobner-subsetp ring-and-order g1 g2) (grobner-subsetp ring-and-order g2 g1)))

(defun grobner-subsetp (ring-and-order g1 g2)
  "Returns T if a list of polynomials G1 generates
an ideal contained in the ideal generated by a polynomial list G2,
both G1 and G2 assumed to be Grobner bases. Returns NIL otherwise."
  (declare (type ring-and-order ring-and-order))
  (every #'(lambda (p) (grobner-member ring-and-order p g2)) g1))

(defun grobner-member (ring-and-order p g)
  "Returns T if a polynomial P belongs to the ideal generated by the
polynomial list G, which is assumed to be a Grobner basis. Returns NIL otherwise."
  (declare (type ring-and-order ring-and-order))
  (poly-zerop (normal-form ring-and-order p g nil)))

;; Calculate F : p^inf
(defun ideal-saturation-1 (ring-and-order f p 
                           &optional 
                             (start 0)
                             (top-reduction-only $poly_top_reduction_only)
                           &aux
                             (ring (ro-ring ring-and-order)))
  "Returns the reduced Grobner basis of the saturation of the ideal
generated by a polynomial list F in the ideal generated by a single
polynomial P. The saturation ideal is defined as the set of
polynomials H such for some natural number n (* (EXPT P N) H) is in
the ideal spanned by F. Geometrically, over an algebraically closed
field, this is the set of polynomials in the ideal generated by F
which do not identically vanish on the variety of P."
  (declare (type ring-and-order ring-and-order))
  (mapcar
   #'poly-contract
   (ring-intersection
    (reduced-grobner
     ring-and-order
     (saturation-extension-1 ring f p)
     start top-reduction-only)
    1)))


;; Calculate F : p1^inf : p2^inf : ... : ps^inf
(defun ideal-polysaturation-1 (ring-and-order f plist 
                               &optional 
                                 (start 0)
                                 (top-reduction-only $poly_top_reduction_only))
  "Returns the reduced Grobner basis of the ideal obtained by a
sequence of successive saturations in the polynomials
of the polynomial list PLIST of the ideal generated by the
polynomial list F."
  (cond
    ((endp plist) 
     (reduced-grobner ring-and-order f start top-reduction-only))
    (t (let ((g (ideal-saturation-1 ring-and-order f (car plist) start top-reduction-only)))
         (ideal-polysaturation-1 ring-and-order g (rest plist) (length g) top-reduction-only)))))

(defun ideal-saturation (ring-and-order f g
                         &optional 
                           (start 0)
                           (top-reduction-only $poly_top_reduction_only)
                         &aux
                           (k (length g))
                           (ring (ro-ring ring-and-order)))
  "Returns the reduced Grobner basis of the saturation of the ideal
generated by a polynomial list F in the ideal generated a polynomial
list G. The saturation ideal is defined as the set of polynomials H
such for some natural number n and some P in the ideal generated by G
the polynomial P**N * H is in the ideal spanned by F.  Geometrically,
over an algebraically closed field, this is the set of polynomials in
the ideal generated by F which do not identically vanish on the
variety of G."
  (declare (type ring-and-order ring-and-order))
  (mapcar
   #'(lambda (q) (poly-contract q k))
   (ring-intersection
    (reduced-grobner ring-and-order
                     (polysaturation-extension ring f g)
                     start
                     top-reduction-only)
    k)))

(defun ideal-polysaturation (ring-and-order f ideal-list
                             &optional 
                               (start 0)
                               (top-reduction-only $poly_top_reduction_only))
  "Returns the reduced Grobner basis of the ideal obtained by a
successive applications of IDEAL-SATURATION to F and lists of
polynomials in the list IDEAL-LIST."
  (declare (type ring-and-order ring-and-order))
  (cond
    ((endp ideal-list) f)
    (t (let ((h (ideal-saturation ring-and-order f (car ideal-list) start top-reduction-only)))
         (ideal-polysaturation ring-and-order h (rest ideal-list) (length h) top-reduction-only)))))