| 1 | \begin{lisp:documentation}{*colored$-$poly$-$debug*}{VARIABLE}{nil }
|
|---|
| 2 | If true debugging output is on.
|
|---|
| 3 | \end{lisp:documentation}
|
|---|
| 4 |
|
|---|
| 5 | \begin{lisp:documentation}{debug$-$cgb}{MACRO}{{\sf \&rest} args }
|
|---|
| 6 | {\ } % NO DOCUMENTATION FOR DEBUG-CGB
|
|---|
| 7 | \end{lisp:documentation}
|
|---|
| 8 |
|
|---|
| 9 | \begin{lisp:documentation}{make$-$colored$-$poly}{FUNCTION}{poly k {\sf \&key} (key \#'identity) (main$-$order \#'lex$>$) (parameter$-$order \#'lex$>$) {\sf \&aux} l }
|
|---|
| 10 | Colored poly is represented as a list
|
|---|
| 11 | (TERM1 TERM2 ... TERMS)
|
|---|
| 12 | where each term is a triple
|
|---|
| 13 | (MONOM . (POLY . COLOR))
|
|---|
| 14 | where monoms and polys have common number of variables while color is
|
|---|
| 15 | one of the three: :RED, :GREEN or :WHITE. This function translates
|
|---|
| 16 | an ordinary polynomial into a colored one by dividing variables into
|
|---|
| 17 | K and N$-$K, where N is the total number of variables in the
|
|---|
| 18 | polynomial poly; the function KEY can be called to select variables
|
|---|
| 19 | in arbitrary order.
|
|---|
| 20 | \end{lisp:documentation}
|
|---|
| 21 |
|
|---|
| 22 | \begin{lisp:documentation}{make$-$colored$-$poly$-$list}{FUNCTION}{plist {\sf \&rest} rest }
|
|---|
| 23 | Translate a list of polynomials PLIST into a list of colored
|
|---|
| 24 | polynomials by calling MAKE$-$COLORED$-$POLY. Returns the resulting
|
|---|
| 25 | list.
|
|---|
| 26 | \end{lisp:documentation}
|
|---|
| 27 |
|
|---|
| 28 | \begin{lisp:documentation}{color$-$poly$-$list}{FUNCTION}{flist {\sf \&optional} (cond (list nil nil)) }
|
|---|
| 29 | Add colors to an ordinary list of polynomials FLIST, according to a
|
|---|
| 30 | condition COND. A condition is a pair of polynomial lists. Each
|
|---|
| 31 | polynomial in COND is a polynomial in parameters only. The list
|
|---|
| 32 | (FIRST COND) is called the ``green list'' and it consists of
|
|---|
| 33 | polynomials which vanish for the parameters associated with the
|
|---|
| 34 | condition. The list (SECOND COND) is called the ``red list
|
|---|
| 35 | \end{lisp:documentation}
|
|---|
| 36 |
|
|---|
| 37 | \begin{lisp:documentation}{color$-$poly}{FUNCTION}{f {\sf \&optional} (cond (list nil nil)) }
|
|---|
| 38 | Add color to a single polynomial F, according to condition COND.
|
|---|
| 39 | See the documentation of COLOR$-$POLY$-$LIST.
|
|---|
| 40 | \end{lisp:documentation}
|
|---|
| 41 |
|
|---|
| 42 | \begin{lisp:documentation}{colored$-$poly$-$to$-$poly}{FUNCTION}{cpoly }
|
|---|
| 43 | For a given colored polynomial CPOLY, removes the colors and
|
|---|
| 44 | it returns the polynomial as an ordinary polynomial with
|
|---|
| 45 | coefficients which are polynomials in parameters.
|
|---|
| 46 | \end{lisp:documentation}
|
|---|
| 47 |
|
|---|
| 48 | \begin{lisp:documentation}{colored$-$poly$-$print}{FUNCTION}{poly vars {\sf \&key} (stream t) (beg t) (print$-$green$-$part nil) (mark$-$coefficients nil) }
|
|---|
| 49 | Print a colored polynomial POLY. Use variables VARS to represent
|
|---|
| 50 | the variables. Some of the variables are going to be used as
|
|---|
| 51 | parameters, according to the length of the monomials in the main
|
|---|
| 52 | monomial and coefficient part of each term in POLY. The key variable
|
|---|
| 53 | STREAM may be used to redirect the output. If parameter
|
|---|
| 54 | PRINT$-$GREEN$-$PART is set then the coefficients which have color
|
|---|
| 55 | :GREEN will be printed, otherwise they are discarded silently. If
|
|---|
| 56 | MARK$-$COEFFICIENTS is not NIL then every coefficient will be marked
|
|---|
| 57 | according to its color, for instance G(U$-$1) would mean that U$-$1
|
|---|
| 58 | is in the green list. Returns P.
|
|---|
| 59 | \end{lisp:documentation}
|
|---|
| 60 |
|
|---|
| 61 | \begin{lisp:documentation}{colored$-$poly$-$print$-$list}{FUNCTION}{poly$-$list vars {\sf \&key} (stream t) (beg t) (print$-$green$-$part nil) (mark$-$coefficients nil) }
|
|---|
| 62 | Pring a list of colored polynomials via a call to
|
|---|
| 63 | COLORED$-$POLY$-$PRINT.
|
|---|
| 64 | \end{lisp:documentation}
|
|---|
| 65 |
|
|---|
| 66 | \begin{lisp:documentation}{determine}{FUNCTION}{f {\sf \&optional} (cond (list nil nil)) (order \#'lex$>$) (ring *coefficient$-$ring*) }
|
|---|
| 67 | This function takes a list of colored polynomials F and a condition
|
|---|
| 68 | COND, and it returns a list of pairs (COND' F') such that COND' cover
|
|---|
| 69 | COND and F' is a ``determined'' version of the colored polynomial
|
|---|
| 70 | list F, i.e. every polynomial has its leading coefficient determined.
|
|---|
| 71 | This means that some of the initial coefficients in each polynomial
|
|---|
| 72 | in F' are in the green list of COND, and the first non$-$green
|
|---|
| 73 | coefficient is in the red list of COND. We note that F' differs from
|
|---|
| 74 | F only by different colors: some of the terms marked :WHITE are now
|
|---|
| 75 | marked either :GREEN or :RED. Coloring is done either by explicitly
|
|---|
| 76 | checking membership in red or green list of COND, or implicitly by
|
|---|
| 77 | performing Grobner basis calculations in the polynomial ring over the
|
|---|
| 78 | parameters. The admissible monomial order ORDER is used only in the
|
|---|
| 79 | parameter space. Also, the ring structure RING is used only for
|
|---|
| 80 | calculations on polynomials of the parameters only.
|
|---|
| 81 | \end{lisp:documentation}
|
|---|
| 82 |
|
|---|
| 83 | \begin{lisp:documentation}{determine$-$1}{FUNCTION}{cond p end gp order ring }
|
|---|
| 84 | Determine a single colored polynomial P according to condition COND.
|
|---|
| 85 | Prepend green part GP to P. Cons the result with END, which should be
|
|---|
| 86 | a list of colored polynomials, and return the resulting list of
|
|---|
| 87 | polynomials. This is an auxillary function of DETERMINE.
|
|---|
| 88 | \end{lisp:documentation}
|
|---|
| 89 |
|
|---|
| 90 | \begin{lisp:documentation}{determine$-$white$-$term}{FUNCTION}{cond term restp end gp order ring }
|
|---|
| 91 | This is an auxillary function of DETERMINE. In this function the
|
|---|
| 92 | parameter COND is a condition. The parameters TERM, RESTP and GP are
|
|---|
| 93 | three parts of a polynomial being processed, where TERM is colored
|
|---|
| 94 | :WHITE. We test the membership in the red and green list of COND we
|
|---|
| 95 | try to determine whether the term is :RED or :GREEN. This is done by
|
|---|
| 96 | performing ideal membership tests in the polynomial ring. Let C be
|
|---|
| 97 | the coefficient of TERM. Thus, C is a polynomial in parameters. We
|
|---|
| 98 | find whether C is in the green list by performing a plain ideal
|
|---|
| 99 | membership test. However, to test properly whether C is in the red
|
|---|
| 100 | list, one needs a different strategy. In fact, we test whether
|
|---|
| 101 | adding C to the red list would produce a non$-$empty set of
|
|---|
| 102 | parameters in some algebraic extension. The test is whether 1 belongs
|
|---|
| 103 | to the saturation ideal of (FIRST COND) in (CONS C (SECOND COND)).
|
|---|
| 104 | Thus, we use POLY$-$SATURATION. If we are successful in determining
|
|---|
| 105 | the color of TERM, we simply change the color of the term and return
|
|---|
| 106 | the list ((COND P)) where P is obtained by appending GP, (LIST TERM)
|
|---|
| 107 | and RESTP. If we cannot determine whether TERM is :RED or :GREEN, we
|
|---|
| 108 | return the list ((COND' P') (COND'' P
|
|---|
| 109 | \end{lisp:documentation}
|
|---|
| 110 |
|
|---|
| 111 | \begin{lisp:documentation}{cond$-$system$-$print}{FUNCTION}{system vars params {\sf \&key} (suppress$-$value t) (print$-$green$-$part nil) (mark$-$coefficients nil) {\sf \&aux} (label 0) }
|
|---|
| 112 | A conditional system SYSTEM is a list of pairs (COND PLIST), where
|
|---|
| 113 | COND is a condition (a pair (GREEN$-$LIST RED$-$LIST)) and PLIST is a
|
|---|
| 114 | list of colored polynomials. This function pretty$-$prints this list
|
|---|
| 115 | of pairs. A conditional system is the data structure returned by
|
|---|
| 116 | GROBNER$-$SYSTEM. This function returns SYSTEM, if SUPPRESS$-$VALUE
|
|---|
| 117 | is non$-$NIL and no value otherwise. If MARK$-$COEFFICIENTS is
|
|---|
| 118 | non$-$NIL coefficients will be marked as in G(u$-$1)*x+R(2)*y, which
|
|---|
| 119 | means that u$-$1 is :GREEN and 2 is :RED.
|
|---|
| 120 | \end{lisp:documentation}
|
|---|
| 121 |
|
|---|
| 122 | \begin{lisp:documentation}{cond$-$print}{FUNCTION}{cond params }
|
|---|
| 123 | Pretty$-$print a condition COND, using symbol list PARAMS as
|
|---|
| 124 | parameter names.
|
|---|
| 125 | \end{lisp:documentation}
|
|---|
| 126 |
|
|---|
| 127 | \begin{lisp:documentation}{add$-$pairs}{FUNCTION}{gs pred }
|
|---|
| 128 | The parameter GS shoud be a Grobner system, i.e. a set of pairs
|
|---|
| 129 | (CONDITION POLY$-$LIST) This functions adds the third component: the
|
|---|
| 130 | list of initial critical pairs (I J), as in the ordinary Grobner
|
|---|
| 131 | basis algorithm. In addition, it adds the length of of the
|
|---|
| 132 | POLY$-$LIST, less 1, as the fourth component. The resulting list of
|
|---|
| 133 | quadruples is returned.
|
|---|
| 134 | \end{lisp:documentation}
|
|---|
| 135 |
|
|---|
| 136 | \begin{lisp:documentation}{cond$-$part}{FUNCTION}{p }
|
|---|
| 137 | Find the part of a colored polynomial P starting with the first
|
|---|
| 138 | non$-$green term.
|
|---|
| 139 | \end{lisp:documentation}
|
|---|
| 140 |
|
|---|
| 141 | \begin{lisp:documentation}{cond$-$hm}{FUNCTION}{p }
|
|---|
| 142 | Return the conditional head monomial of a colored polynomial P.
|
|---|
| 143 | \end{lisp:documentation}
|
|---|
| 144 |
|
|---|
| 145 | \begin{lisp:documentation}{delete$-$green$-$polys}{FUNCTION}{gamma }
|
|---|
| 146 | Delete totally green polynomials from in a grobner system GAMMA.
|
|---|
| 147 | \end{lisp:documentation}
|
|---|
| 148 |
|
|---|
| 149 | \begin{lisp:documentation}{grobner$-$system}{FUNCTION}{f {\sf \&key} (cover (list '(nil nil))) (main$-$order \#'lex$>$) (parameter$-$order \#'lex$>$) (reduce t) (green$-$reduce t) (top$-$reduction$-$only
|
|---|
| 150 | nil) (ring
|
|---|
| 151 | *coefficient$-$ring*) {\sf \&aux} (cover
|
|---|
| 152 | (saturate$-$cover
|
|---|
| 153 | cover
|
|---|
| 154 | parameter$-$order
|
|---|
| 155 | ring)) (gamma
|
|---|
| 156 | (delete$-$green$-$polys
|
|---|
| 157 | (mapcan
|
|---|
| 158 | \#'(lambda (cond) (determine f cond parameter$-$order ring))
|
|---|
| 159 | cover))) }
|
|---|
| 160 | This function returns a grobner system, given a list of colored
|
|---|
| 161 | polynomials F, Other parameters are:
|
|---|
| 162 | A cover COVER, i.e. a list of conditions, i.e. pairs of the form
|
|---|
| 163 | (GREEN$-$LIST RED$-$LIST), where GREEN$-$LIST and RED$-$LIST are to
|
|---|
| 164 | lists of ordinary polynomials in parameters. A monomial order
|
|---|
| 165 | MAIN$-$ORDER used on main variables (not parameters). A monomial
|
|---|
| 166 | order PARAMETER$-$ORDER used in calculations with parameters only.
|
|---|
| 167 | REDUCE, a flag deciding whether COLORED$-$REDUCTION will be performed
|
|---|
| 168 | on the resulting grobner system. GREEN$-$REDUCE, a flag deciding
|
|---|
| 169 | whether the green list of each condition will be reduced in a form of
|
|---|
| 170 | a reduced Grobner basis. TOP$-$REDUCTION$-$ONLY, a flag deciding
|
|---|
| 171 | whether in the internal calculations in the space of parameters top
|
|---|
| 172 | reduction only will be used. RING, a structure as in the package
|
|---|
| 173 | COEFFICIENT$-$RING, used in operations on the coefficients of the
|
|---|
| 174 | polynomials in parameters.
|
|---|
| 175 | \end{lisp:documentation}
|
|---|
| 176 |
|
|---|
| 177 | \begin{lisp:documentation}{reorder$-$pairs}{FUNCTION}{b bnew g pred {\sf \&optional} (sort$-$first nil) }
|
|---|
| 178 | Reorder pairs according to some heuristic. The heuristic at this time
|
|---|
| 179 | is ad hoc, in the future it should be replaced with sugar strategy
|
|---|
| 180 | and a mechanism for implementing new heuristic strategies, as in the
|
|---|
| 181 | GROBNER package.
|
|---|
| 182 | \end{lisp:documentation}
|
|---|
| 183 |
|
|---|
| 184 | \begin{lisp:documentation}{colored$-$criterion$-$1}{FUNCTION}{i j f }
|
|---|
| 185 | Buchberger criterion 1 for colored polynomials.
|
|---|
| 186 | \end{lisp:documentation}
|
|---|
| 187 |
|
|---|
| 188 | \begin{lisp:documentation}{colored$-$criterion$-$2}{FUNCTION}{i j f b s }
|
|---|
| 189 | Buchberger criterion 2 for colored polynomials.
|
|---|
| 190 | \end{lisp:documentation}
|
|---|
| 191 |
|
|---|
| 192 | \begin{lisp:documentation}{cond$-$normal$-$form}{FUNCTION}{f fl main$-$order parameter$-$order top$-$reduction$-$only ring }
|
|---|
| 193 | Returns the conditional normal form of a colored polynomial F with
|
|---|
| 194 | respect to the list of colored polynomials FL. The list FL is assumed
|
|---|
| 195 | to consist of determined polynomials, i.e. such that the first term
|
|---|
| 196 | which is not marked :GREEN is :RED.
|
|---|
| 197 | \end{lisp:documentation}
|
|---|
| 198 |
|
|---|
| 199 | \begin{lisp:documentation}{cond$-$spoly}{FUNCTION}{f g main$-$order parameter$-$order ring }
|
|---|
| 200 | Returns the conditional S$-$polynomial of two colored polynomials F
|
|---|
| 201 | and G. Both polynomials are assumed to be determined.
|
|---|
| 202 | \end{lisp:documentation}
|
|---|
| 203 |
|
|---|
| 204 | \begin{lisp:documentation}{cond$-$lm}{FUNCTION}{f }
|
|---|
| 205 | Returns the conditional leading monomial of a colored polynomial F,
|
|---|
| 206 | which is assumed to be determined.
|
|---|
| 207 | \end{lisp:documentation}
|
|---|
| 208 |
|
|---|
| 209 | \begin{lisp:documentation}{cond$-$lc}{FUNCTION}{f }
|
|---|
| 210 | Returns the conditional leading coefficient of a colored polynomial
|
|---|
| 211 | F, which is assumed to be determined.
|
|---|
| 212 | \end{lisp:documentation}
|
|---|
| 213 |
|
|---|
| 214 | \begin{lisp:documentation}{colored$-$term$-$times$-$poly}{FUNCTION}{term f order ring }
|
|---|
| 215 | Returns the product of a colored term TERM and a colored polynomial
|
|---|
| 216 | F.
|
|---|
| 217 | \end{lisp:documentation}
|
|---|
| 218 |
|
|---|
| 219 | \begin{lisp:documentation}{colored$-$scalar$-$times$-$poly}{FUNCTION}{c f ring }
|
|---|
| 220 | Returns the product of an element of the coefficient ring C a colored
|
|---|
| 221 | polynomial F.
|
|---|
| 222 | \end{lisp:documentation}
|
|---|
| 223 |
|
|---|
| 224 | \begin{lisp:documentation}{colored$-$term*}{FUNCTION}{term1 term2 order ring }
|
|---|
| 225 | Returns the product of two colored terms TERM1 and TERM2.
|
|---|
| 226 | \end{lisp:documentation}
|
|---|
| 227 |
|
|---|
| 228 | \begin{lisp:documentation}{color*}{FUNCTION}{c1 c2 }
|
|---|
| 229 | Returns a product of two colores. Rules:
|
|---|
| 230 | :red * :red yields :red
|
|---|
| 231 | any * :green yields :green
|
|---|
| 232 | otherwise the result is :white.
|
|---|
| 233 | \end{lisp:documentation}
|
|---|
| 234 |
|
|---|
| 235 | \begin{lisp:documentation}{color+}{FUNCTION}{c1 c2 }
|
|---|
| 236 | Returns a sum of colors. Rules:
|
|---|
| 237 | :green + :green yields :green,
|
|---|
| 238 | :red + :green yields :red
|
|---|
| 239 | any other result is :white.
|
|---|
| 240 | \end{lisp:documentation}
|
|---|
| 241 |
|
|---|
| 242 | \begin{lisp:documentation}{color$-$}{FUNCTION}{c1 c2 }
|
|---|
| 243 | Identical to COLOR+.
|
|---|
| 244 | \end{lisp:documentation}
|
|---|
| 245 |
|
|---|
| 246 | \begin{lisp:documentation}{colored$-$poly+}{FUNCTION}{p q main$-$order parameter$-$order ring }
|
|---|
| 247 | Returns the sum of colored polynomials P and Q.
|
|---|
| 248 | \end{lisp:documentation}
|
|---|
| 249 |
|
|---|
| 250 | \begin{lisp:documentation}{colored$-$poly$-$}{FUNCTION}{p q main$-$order parameter$-$order ring }
|
|---|
| 251 | Returns the difference of colored polynomials P and Q.
|
|---|
| 252 | \end{lisp:documentation}
|
|---|
| 253 |
|
|---|
| 254 | \begin{lisp:documentation}{colored$-$term$-$uminus}{FUNCTION}{term ring }
|
|---|
| 255 | Returns the negation of a colored term TERM.
|
|---|
| 256 | \end{lisp:documentation}
|
|---|
| 257 |
|
|---|
| 258 | \begin{lisp:documentation}{colored$-$minus$-$poly}{FUNCTION}{p ring }
|
|---|
| 259 | Returns the negation of a colored polynomial P.
|
|---|
| 260 | \end{lisp:documentation}
|
|---|
| 261 |
|
|---|
| 262 | \begin{lisp:documentation}{string$-$grobner$-$system}{FUNCTION}{f vars params {\sf \&key} (cover (list (list "[]" "[]"))) (main$-$order \#'lex$>$) (parameter$-$order \#'lex$>$) (ring
|
|---|
| 263 | *coefficient$-$ring*) (suppress$-$value
|
|---|
| 264 | t) (suppress$-$printing
|
|---|
| 265 | nil) (mark$-$coefficients
|
|---|
| 266 | nil) (reduce
|
|---|
| 267 | t) (green$-$reduce
|
|---|
| 268 | t) {\sf \&aux} (f
|
|---|
| 269 | (parse$-$to$-$colored$-$poly$-$list
|
|---|
| 270 | f
|
|---|
| 271 | vars
|
|---|
| 272 | params
|
|---|
| 273 | main$-$order
|
|---|
| 274 | parameter$-$order)) (cover
|
|---|
| 275 | (string$-$cover
|
|---|
| 276 | cover
|
|---|
| 277 | params
|
|---|
| 278 | parameter$-$order)) }
|
|---|
| 279 | An interface to GROBNER$-$SYSTEM in which polynomials can be
|
|---|
| 280 | specified in infix notations as strings. Lists of polynomials are
|
|---|
| 281 | comma$-$separated list marked by a matchfix operators []
|
|---|
| 282 | \end{lisp:documentation}
|
|---|
| 283 |
|
|---|
| 284 | \begin{lisp:documentation}{string$-$cond}{FUNCTION}{cond params {\sf \&optional} (order \#'lex$>$) }
|
|---|
| 285 | Return the internal representation of a condition COND, specified
|
|---|
| 286 | as pairs of strings (GREEN$-$LIST RED$-$LIST). GREEN$-$LIST and
|
|---|
| 287 | RED$-$LIST in the input are assumed to be strings which parse to two
|
|---|
| 288 | lists of polynomials with respect to variables whose names are in the
|
|---|
| 289 | list of symbols PARAMS. ORDER is the predicate used to sort the terms
|
|---|
| 290 | of the polynomials.
|
|---|
| 291 | \end{lisp:documentation}
|
|---|
| 292 |
|
|---|
| 293 | \begin{lisp:documentation}{string$-$cover}{FUNCTION}{cover params {\sf \&optional} (order \#'lex$>$) }
|
|---|
| 294 | Returns the internal representation of COVER, given in the form of
|
|---|
| 295 | a list of conditions. See STRING$-$COND for description of a
|
|---|
| 296 | condition.
|
|---|
| 297 | \end{lisp:documentation}
|
|---|
| 298 |
|
|---|
| 299 | \begin{lisp:documentation}{saturate$-$cover}{FUNCTION}{cover order ring }
|
|---|
| 300 | Brings every condition of a list of conditions COVER to the form (G
|
|---|
| 301 | R) where G is saturated with respect to R and G is a Grobner basis
|
|---|
| 302 | We could reduce R so that the elements of R are relatively prime,
|
|---|
| 303 | but this is not currently done.
|
|---|
| 304 | \end{lisp:documentation}
|
|---|
| 305 |
|
|---|
| 306 | \begin{lisp:documentation}{saturate$-$cond}{FUNCTION}{cond order ring }
|
|---|
| 307 | Saturate a single condition COND. An auxillary function of
|
|---|
| 308 | SATURATE$-$COVER.
|
|---|
| 309 | \end{lisp:documentation}
|
|---|
| 310 |
|
|---|
| 311 | \begin{lisp:documentation}{string$-$determine}{FUNCTION}{f vars params {\sf \&key} (cond '([] [])) (main$-$order \#'lex$>$) (parameter$-$order \#'lex$>$) (suppress$-$value t) (suppress$-$printing
|
|---|
| 312 | nil) (mark$-$coefficients
|
|---|
| 313 | nil) (ring
|
|---|
| 314 | *coefficient$-$ring*) {\sf \&aux} (f
|
|---|
| 315 | (parse$-$to$-$colored$-$poly$-$list
|
|---|
| 316 | f
|
|---|
| 317 | vars
|
|---|
| 318 | params
|
|---|
| 319 | main$-$order
|
|---|
| 320 | parameter$-$order)) (cond
|
|---|
| 321 | (string$-$cond
|
|---|
| 322 | cond
|
|---|
| 323 | params
|
|---|
| 324 | parameter$-$order)) }
|
|---|
| 325 | A string interface to DETERMINE. See the documentation of
|
|---|
| 326 | STRING$-$GROBNER$-$SYSTEM.
|
|---|
| 327 | \end{lisp:documentation}
|
|---|
| 328 |
|
|---|
| 329 | \begin{lisp:documentation}{tidy$-$grobner$-$system}{FUNCTION}{gs main$-$order parameter$-$order reduce green$-$reduce ring }
|
|---|
| 330 | Apply TIDY$-$PAIR to every pair of a Grobner system.
|
|---|
| 331 | \end{lisp:documentation}
|
|---|
| 332 |
|
|---|
| 333 | \begin{lisp:documentation}{tidy$-$pair}{FUNCTION}{pair main$-$order parameter$-$order reduce green$-$reduce ring {\sf \&aux} gs }
|
|---|
| 334 | Make the output of Grobner system more readable by performing
|
|---|
| 335 | certain simplifications on an element of a Grobner system.
|
|---|
| 336 | If REDUCE is non$-$NIL then COLORED$-$reduction will be performed.
|
|---|
| 337 | In addition TIDY$-$COND is called on the condition part of the pair
|
|---|
| 338 | PAIR.
|
|---|
| 339 | \end{lisp:documentation}
|
|---|
| 340 |
|
|---|
| 341 | \begin{lisp:documentation}{tidy$-$cond}{FUNCTION}{cond order ring }
|
|---|
| 342 | Currently saturates condition COND and does RED$-$REDUCTION on the
|
|---|
| 343 | red list.
|
|---|
| 344 | \end{lisp:documentation}
|
|---|
| 345 |
|
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| 346 | \begin{lisp:documentation}{colored$-$reduction}{FUNCTION}{cond p main$-$order parameter$-$order ring {\sf \&aux} (open (list (list cond nil p))) closed }
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| 347 | Reduce a list of colored polynomials P. The difficulty as compared
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| 348 | to the usual Buchberger algorithm is that the polys may have the same
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| 349 | leading monomial which may result in cancellations and polynomials
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| 350 | which may not be determined. Thus, when we find those, we will have
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| 351 | to split the condition by calling determine. Returns a list of pairs
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| 352 | (COND' P') where P' is a reduced grobner basis with respect to any
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| 353 | parameter choice compatible with condition COND'. Moreover, COND'
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| 354 | form a cover of COND.
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| 355 | \end{lisp:documentation}
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| 356 |
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| 357 | \begin{lisp:documentation}{green$-$reduce$-$colored$-$poly}{FUNCTION}{cond f parameter$-$order ring }
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| 358 | It takes a colored polynomial F and it returns a modified
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| 359 | polynomial obtained by reducing coefficient of F modulo green list of
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| 360 | the condition COND.
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| 361 | \end{lisp:documentation}
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| 362 |
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| 363 | \begin{lisp:documentation}{green$-$reduce$-$colored$-$list}{FUNCTION}{cond fl parameter$-$order ring }
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| 364 | Apply GREEN$-$REDUCE$-$COLORED$-$POLY to a list of polynomials FL.
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| 365 | \end{lisp:documentation}
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| 366 |
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| 367 | \begin{lisp:documentation}{cond$-$system$-$green$-$reduce}{FUNCTION}{gs parameter$-$order ring }
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| 368 | Apply GREEN$-$REDUCE$-$COLORED$-$LIST to every pair of
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| 369 | a grobner system GS.
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| 370 | \end{lisp:documentation}
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| 371 |
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| 372 | \begin{lisp:documentation}{parse$-$to$-$colored$-$poly$-$list}{FUNCTION}{f vars params main$-$order parameter$-$order {\sf \&aux} (k (length vars)) (vars$-$params (append vars params)) }
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| 373 | Parse a list of polynomials F, given as a string, with respect to
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| 374 | a list of variables VARS, given as a list of symbols, to the internal
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| 375 | representation of a colored polynomial. The polynomials will be
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| 376 | properly sorted by MAIN$-$ORDER, with the coefficients, which are
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| 377 | polynomials in parameters, sorted by PARAMETER$-$ORDER. Both orders
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| 378 | must be admissible monomial orders. This form is suitable for parsing
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| 379 | polynomials with integer coefficients.
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| 380 | \end{lisp:documentation}
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| 381 |
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| 382 | \begin{lisp:documentation}{red$-$reduction}{FUNCTION}{p pred ring {\sf \&aux} (p (remove$-$if \#'poly$-$constant$-$p p)) }
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| 383 | Takes a family of polynomials and produce a list whose prime factors
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| 384 | are the same but they are relatively prime
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| 385 | Repetitively used the following procedure: it finds two elements f, g
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| 386 | of P which are not relatively prime; it replaces f and g with
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| 387 | f/GCD(f,g), g/ GCD(f,f) and GCD(f,g).
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| 388 | \end{lisp:documentation}
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| 389 |
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