source: branches/f4grobner/ideal.lisp@ 67

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;; Operations in ideal theory
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;; Does the term depend on variable K?
(defun term-depends-p (term k)
  "Return T if the term TERM depends on variable number K."
  (monom-depends-p (term-monom term) k))

;; 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 is a Grobner basis
and it calculates the intersection with the ring R[x[k+1],...,x[n]], i.e.
it discards polynomials which depend on variables x[0], x[1], ..., x[k]."
  (dotimes (i k plist)
    (setf plist
          (remove-if #'(lambda (p)
                         (poly-depends-p p i))
                     plist))))

(defun elimination-ideal (ring flist k
                          &optional (top-reduction-only $poly_top_reduction_only) (start 0)
                          &aux (*monomial-order*
                                (or *elimination-order*
                                    (elimination-order k))))
  (ring-intersection (reduced-grobner ring flist start top-reduction-only) k))

(defun colon-ideal (ring 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 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."
  (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
                (list (make-term
                       (make-monom (monom-dimension (poly-lm (find-if-not #'poly-zerop f)))
                                   :initial-element 0)
                       (funcall (ring-unit ring))))))))
    ((endp (cdr g))
     (colon-ideal-1 ring f (car g) top-reduction-only))
    (t
     (ideal-intersection ring
                         (colon-ideal-1 ring f (car g) top-reduction-only)
                         (colon-ideal ring f (rest g) top-reduction-only)
                         top-reduction-only))))

(defun colon-ideal-1 (ring 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."
  (mapcar #'(lambda (x) (poly-exact-divide ring x g)) (ideal-intersection ring f (list g) top-reduction-only)))


(defun ideal-intersection (ring f g &optional (top-reduction-only $poly_top_reduction_only)
                           &aux (*monomial-order* (or *elimination-order*
                                                      #'elimination-order-1)))
  (mapcar #'poly-contract
          (ring-intersection
           (reduced-grobner
            ring
            (append (mapcar #'(lambda (p) (poly-extend p (make-monom 1 :initial-element 1))) f)
                    (mapcar #'(lambda (p)
                                (poly-append (poly-extend (poly-uminus ring p)
                                                          (make-monom 1 :initial-element 1))
                                             (poly-extend p)))
                            g))
            0
            top-reduction-only)
           1)))

(defun poly-lcm (ring f g)
  "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 m (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 (list f) (list g) nil)))))))))

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

(defun grobner-subsetp (ring 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."
  (every #'(lambda (p) (grobner-member ring p g2)) g1))

(defun grobner-member (ring 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."
  (poly-zerop (normal-form ring p g nil)))

;; Calculate F : p^inf
(defun ideal-saturation-1 (ring f p start &optional (top-reduction-only $poly_top_reduction_only)
                           &aux (*monomial-order* (or *elimination-order*
                                                      #'elimination-order-1)))
  "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
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."
  (mapcar
   #'poly-contract
   (ring-intersection
    (reduced-grobner
     ring
     (saturation-extension-1 ring f p)
     start top-reduction-only)
    1)))



;; Calculate F : p1^inf : p2^inf : ... : ps^inf
(defun ideal-polysaturation-1 (ring f plist start &optional (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 f start top-reduction-only))
   (t (let ((g (ideal-saturation-1 ring f (car plist) start top-reduction-only)))
        (ideal-polysaturation-1 ring g (rest plist) (length g) top-reduction-only)))))

(defun ideal-saturation (ring f g start &optional (top-reduction-only $poly_top_reduction_only)
                         &aux
                         (k (length g))
                         (*monomial-order* (or *elimination-order*
                                               (elimination-order k))))
  "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."
  (mapcar
   #'(lambda (q) (poly-contract q k))
   (ring-intersection
    (reduced-grobner ring
                     (polysaturation-extension ring f g)
                     start
                     top-reduction-only)
    k)))

(defun ideal-polysaturation (ring f ideal-list start &optional (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."
  (cond
   ((endp ideal-list) f)
   (t (let ((h (ideal-saturation ring f (car ideal-list) start top-reduction-only)))
        (ideal-polysaturation ring h (rest ideal-list) (length h) top-reduction-only)))))