Hilbert series #2407
JohnAAbbott
started this conversation in
General
Hilbert series
#2407
Replies: 9 comments 1 reply
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The denominator depends only on the graded ring, not on the conretely given
ideal or module. It carries no extra information which is sensitive to the user and
is just used, together with the more interesting numerator, to compute the
Hilbert function. So I do not see any reason why the user should take a look
at it. Also, what data type do you suggest? A polynomial is a sum of terms
which will be processed when computing the Hilbert function. But what is
a product of polynomials (which would be expanded anyway when computing
the Hilbert function)?
… Am 24.05.2023 um 12:38 schrieb JohnAAbbott ***@***.***>:
The hilbert series we compute are returned as a pair of polynomials (effectively numerator and denominator, but we may want to allow common factors). The denominator is a product of factors of the form (1-t^d); an easy generalization applies when we have a grading over ZZ^k.
Currently Oscar multiplies the denominator factors together producing a potentially large product (which may be expensive to factorize). In contrast, CoCoA leaves the denominator as a product of simple factors. Should Oscar also leave the denominator as a product of simple factors?
Here is a (contrived) concrete example: CoCoA produces the following result
(1 - t^1060) / ( (1-t^2)*(1-t^3)*(1-t^5)*(1-t^7)*(1-t^11)*(1-t^13)*(1-t^17)*(1-t^19)*(1-t^23)*(1-t^29)*(1-t^31)*(1-t^37)*(1-t^41)*(1-t^43)*(1-t^47)*(1-t^53)*(1-t^59)*(1-t^61)*(1-t^67)*(1-t^71)*(1-t^73)*(1-t^79)*(1-t^83)*(1-t^89)*(1-t^97) )
In contrast Oscar produces the equivalent result:
(-t^1060 + 1, -t^1060 + t^1058 + t^1057 - t^1052 - t^1051 + t^1049 - t^1044 + t^1041 + t^1037 - t^1036 - t^1035 + t^1033 + t^1031 + t^1029 - t^1028 - t^1027 - t^1026 + t^1025 - t^1024 + t^1023 + t^1022 + t^1021 - 2*t^1020 - t^1018 + t^1017 - t^1016 + 3*t^1013 - 2*t^1012 - t^1011 - t^1010 + 2*t^1009 - t^1008 + 2*t^1007 - t^1006 + 2*t^1005 - t^1004 - 2*t^1002 + 3*t^1001 - 3*t^1000 + t^999 + 2*t^997 - 4*t^996 + t^995 - 2*t^994 + 4*t^993 - t^992 + t^991 - 3*t^990 + 5*t^989 - 4*t^988 + t^987 - 2*t^986 + 3*t^985 - 4*t^984 + 4*t^983 - 3*t^982 + 5*t^981 - 3*t^980 + t^979 - 4*t^978 + 8*t^977 - 6*t^976 + t^975 - 2*t^974 + 7*t^973 - 6*t^972 + 4*t^971 - 8*t^970 + 6*t^969 - 3*t^968 + 4*t^967 - 6*t^966 + 10*t^965 - 8*t^964 + 4*t^963 - 5*t^962 + 6*t^961 - 10*t^960 + 9*t^959 - 7*t^958 + 10*t^957 - 6*t^956 + 5*t^955 - 9*t^954 + 14*t^953 - 12*t^952 + 5*t^951 - 7*t^950 + 12*t^949 - 12*t^948 + 11*t^947 - 13*t^946 + 12*t^945 - 9*t^944 + 8*t^943 - 13*t^942 + 19*t^941 - 17*t^940 + 9*t^939 - 8*t^938 + 16*t^937 - 18*t^936 + 14*t^935 - 18*t^934 + 18*t^933 - 12*t^932 + 12*t^931 - 19*t^930 + 27*t^929 - 19*t^928 + 14*t^927 - 16*t^926 + 18*t^925 - 23*t^924 + 24*t^923 - 22*t^922 + 22*t^921 - 20*t^920 + 17*t^919 - 24*t^918 + 34*t^917 - 30*t^916 + 17*t^915 - 19*t^914 + 29*t^913 - 31*t^912 + 29*t^911 - 32*t^910 + 31*t^909 - 24*t^908 + 23*t^907 - 34*t^906 + 43*t^905 - 36*t^904 + 28*t^903 - 26*t^902 + 36*t^901 - 42*t^900 + 39*t^899 - 41*t^898 + 40*t^897 - 33*t^896 + 30*t^895 - 44*t^894 + 57*t^893 - 47*t^892 + 34*t^891 - 38*t^890 + 47*t^889 - 51*t^888 + 52*t^887 - 53*t^886 + 49*t^885 - 44*t^884 + 43*t^883 - 55*t^882 + 69*t^881 - 63*t^880 + 47*t^879 - 44*t^878 + 61*t^877 - 68*t^876 + 63*t^875 - 65*t^874 + 68*t^873 - 55*t^872 + 51*t^871 - 75*t^870 + 87*t^869 - 74*t^868 + 60*t^867 - 61*t^866 + 74*t^865 - 81*t^864 + 82*t^863 - 84*t^862 + 80*t^861 - 72*t^860 + 69*t^859 - 88*t^858 + 107*t^857 - 95*t^856 + 73*t^855 - 75*t^854 + 95*t^853 - 100*t^852 + 100*t^851 - 104*t^850 + 99*t^849 - 86*t^848 + 88*t^847 - 110*t^846 + 125*t^845 - 117*t^844 + 94*t^843 - 92*t^842 + 114*t^841 - 125*t^840 + 122*t^839 - 122*t^838 + 123*t^837 - 109*t^836 + 101*t^835 - 133*t^834 + 156*t^833 - 137*t^832 + 112*t^831 - 117*t^830 + 139*t^829 - 144*t^828 + 147*t^827 - 151*t^826 + 143*t^825 - 127*t^824 + 131*t^823 - 158*t^822 + 178*t^821 - 168*t^820 + 137*t^819 - 136*t^818 + 164*t^817 - 174*t^816 + 172*t^815 - 176*t^814 + 172*t^813 - 153*t^812 + 152*t^811 - 188*t^810 + 211*t^809 - 192*t^808 + 162*t^807 - 165*t^806 + 189*t^805 - 203*t^804 + 205*t^803 - 202*t^802 + 201*t^801 - 184*t^800 + 178*t^799 - 213*t^798 + 246*t^797 - 225*t^796 + 184*t^795 - 193*t^794 + 224*t^793 - 230*t^792 + 233*t^791 - 241*t^790 + 230*t^789 - 207*t^788 + 210*t^787 - 250*t^786 + 274*t^785 - 254*t^784 + 221*t^783 - 220*t^782 + 250*t^781 - 270*t^780 + 266*t^779 - 266*t^778 + 265*t^777 - 239*t^776 + 237*t^775 - 279*t^774 + 315*t^773 - 290*t^772 + 244*t^771 - 255*t^770 + 286*t^769 - 297*t^768 + 301*t^767 - 301*t^766 + 294*t^765 - 272*t^764 + 268*t^763 - 313*t^762 + 349*t^761 - 323*t^760 + 278*t^759 - 282*t^758 + 320*t^757 - 334*t^756 + 327*t^755 - 337*t^754 + 332*t^753 - 296*t^752 + 299*t^751 - 351*t^750 + 384*t^749 - 351*t^748 + 311*t^747 - 316*t^746 + 344*t^745 - 368*t^744 + 367*t^743 - 364*t^742 + 360*t^741 - 335*t^740 + 328*t^739 - 375*t^738 + 419*t^737 - 389*t^736 + 333*t^735 - 342*t^734 + 385*t^733 - 396*t^732 + 391*t^731 - 401*t^730 + 393*t^729 - 356*t^728 + 357*t^727 - 409*t^726 + 447*t^725 - 415*t^724 + 364*t^723 - 369*t^722 + 411*t^721 - 427*t^720 + 418*t^719 - 426*t^718 + 422*t^717 - 384*t^716 + 377*t^715 - 437*t^714 + 478*t^713 - 437*t^712 + 389*t^711 - 397*t^710 + 432*t^709 - 450*t^708 + 445*t^707 - 450*t^706 + 440*t^705 - 406*t^704 + 406*t^703 - 453*t^702 + 498*t^701 - 468*t^700 + 405*t^699 - 410*t^698 + 461*t^697 - 470*t^696 + 456*t^695 - 469*t^694 + 466*t^693 - 423*t^692 + 414*t^691 - 478*t^690 + 518*t^689 - 475*t^688 + 421*t^687 - 430*t^686 + 471*t^685 - 482*t^684 + 474*t^683 - 482*t^682 + 474*t^681 - 438*t^680 + 428*t^679 - 482*t^678 + 531*t^677 - 487*t^676 + 427*t^675 - 439*t^674 + 482*t^673 - 487*t^672 + 479*t^671 - 491*t^670 + 480*t^669 - 441*t^668 + 434*t^667 - 486*t^666 + 529*t^665 - 493*t^664 + 430*t^663 - 436*t^662 + 486*t^661 - 494*t^660 + 472*t^659 - 484*t^658 + 487*t^657 - 440*t^656 + 422*t^655 - 485*t^654 + 531*t^653 - 481*t^652 + 422*t^651 - 438*t^650 + 480*t^649 - 475*t^648 + 468*t^647 - 479*t^646 + 470*t^645 - 433*t^644 + 416*t^643 - 470*t^642 + 515*t^641 - 470*t^640 + 410*t^639 - 422*t^638 + 464*t^637 - 465*t^636 + 444*t^635 - 458*t^634 + 459*t^633 - 414*t^632 + 396*t^631 - 450*t^630 + 493*t^629 - 448*t^628 + 388*t^627 - 399*t^626 + 443*t^625 - 440*t^624 + 418*t^623 - 427*t^622 + 437*t^621 - 393*t^620 + 363*t^619 - 421*t^618 + 466*t^617 - 414*t^616 + 353*t^615 - 375*t^614 + 416*t^613 - 400*t^612 + 380*t^611 - 401*t^610 + 400*t^609 - 354*t^608 + 334*t^607 - 384*t^606 + 420*t^605 - 377*t^604 + 322*t^603 - 333*t^602 + 375*t^601 - 364*t^600 + 336*t^599 - 350*t^598 + 363*t^597 - 317*t^596 + 287*t^595 - 337*t^594 + 376*t^593 - 329*t^592 + 274*t^591 - 292*t^590 + 329*t^589 - 315*t^588 + 283*t^587 - 300*t^586 + 316*t^585 - 269*t^584 + 236*t^583 - 285*t^582 + 322*t^581 - 276*t^580 + 219*t^579 - 242*t^578 + 283*t^577 - 251*t^576 + 223*t^575 - 249*t^574 + 260*t^573 - 213*t^572 + 185*t^571 - 227*t^570 + 257*t^569 - 214*t^568 + 168*t^567 - 185*t^566 + 218*t^565 - 197*t^564 + 160*t^563 - 181*t^562 + 201*t^561 - 159*t^560 + 123*t^559 - 159*t^558 + 194*t^557 - 153*t^556 + 100*t^555 - 126*t^554 + 162*t^553 - 126*t^552 + 91*t^551 - 116*t^550 + 140*t^549 - 96*t^548 + 58*t^547 - 93*t^546 + 125*t^545 - 82*t^544 + 36*t^543 - 64*t^542 + 98*t^541 - 57*t^540 + 17*t^539 - 51*t^538 + 73*t^537 - 29*t^536 - 5*t^535 - 26*t^534 + 49*t^533 - 12*t^532 - 26*t^531 + 26*t^529 + 12*t^528 - 49*t^527 + 26*t^526 + 5*t^525 + 29*t^524 - 73*t^523 + 51*t^522 - 17*t^521 + 57*t^520 - 98*t^519 + 64*t^518 - 36*t^517 + 82*t^516 - 125*t^515 + 93*t^514 - 58*t^513 + 96*t^512 - 140*t^511 + 116*t^510 - 91*t^509 + 126*t^508 - 162*t^507 + 126*t^506 - 100*t^505 + 153*t^504 - 194*t^503 + 159*t^502 - 123*t^501 + 159*t^500 - 201*t^499 + 181*t^498 - 160*t^497 + 197*t^496 - 218*t^495 + 185*t^494 - 168*t^493 + 214*t^492 - 257*t^491 + 227*t^490 - 185*t^489 + 213*t^488 - 260*t^487 + 249*t^486 - 223*t^485 + 251*t^484 - 283*t^483 + 242*t^482 - 219*t^481 + 276*t^480 - 322*t^479 + 285*t^478 - 236*t^477 + 269*t^476 - 316*t^475 + 300*t^474 - 283*t^473 + 315*t^472 - 329*t^471 + 292*t^470 - 274*t^469 + 329*t^468 - 376*t^467 + 337*t^466 - 287*t^465 + 317*t^464 - 363*t^463 + 350*t^462 - 336*t^461 + 364*t^460 - 375*t^459 + 333*t^458 - 322*t^457 + 377*t^456 - 420*t^455 + 384*t^454 - 334*t^453 + 354*t^452 - 400*t^451 + 401*t^450 - 380*t^449 + 400*t^448 - 416*t^447 + 375*t^446 - 353*t^445 + 414*t^444 - 466*t^443 + 421*t^442 - 363*t^441 + 393*t^440 - 437*t^439 + 427*t^438 - 418*t^437 + 440*t^436 - 443*t^435 + 399*t^434 - 388*t^433 + 448*t^432 - 493*t^431 + 450*t^430 - 396*t^429 + 414*t^428 - 459*t^427 + 458*t^426 - 444*t^425 + 465*t^424 - 464*t^423 + 422*t^422 - 410*t^421 + 470*t^420 - 515*t^419 + 470*t^418 - 416*t^417 + 433*t^416 - 470*t^415 + 479*t^414 - 468*t^413 + 475*t^412 - 480*t^411 + 438*t^410 - 422*t^409 + 481*t^408 - 531*t^407 + 485*t^406 - 422*t^405 + 440*t^404 - 487*t^403 + 484*t^402 - 472*t^401 + 494*t^400 - 486*t^399 + 436*t^398 - 430*t^397 + 493*t^396 - 529*t^395 + 486*t^394 - 434*t^393 + 441*t^392 - 480*t^391 + 491*t^390 - 479*t^389 + 487*t^388 - 482*t^387 + 439*t^386 - 427*t^385 + 487*t^384 - 531*t^383 + 482*t^382 - 428*t^381 + 438*t^380 - 474*t^379 + 482*t^378 - 474*t^377 + 482*t^376 - 471*t^375 + 430*t^374 - 421*t^373 + 475*t^372 - 518*t^371 + 478*t^370 - 414*t^369 + 423*t^368 - 466*t^367 + 469*t^366 - 456*t^365 + 470*t^364 - 461*t^363 + 410*t^362 - 405*t^361 + 468*t^360 - 498*t^359 + 453*t^358 - 406*t^357 + 406*t^356 - 440*t^355 + 450*t^354 - 445*t^353 + 450*t^352 - 432*t^351 + 397*t^350 - 389*t^349 + 437*t^348 - 478*t^347 + 437*t^346 - 377*t^345 + 384*t^344 - 422*t^343 + 426*t^342 - 418*t^341 + 427*t^340 - 411*t^339 + 369*t^338 - 364*t^337 + 415*t^336 - 447*t^335 + 409*t^334 - 357*t^333 + 356*t^332 - 393*t^331 + 401*t^330 - 391*t^329 + 396*t^328 - 385*t^327 + 342*t^326 - 333*t^325 + 389*t^324 - 419*t^323 + 375*t^322 - 328*t^321 + 335*t^320 - 360*t^319 + 364*t^318 - 367*t^317 + 368*t^316 - 344*t^315 + 316*t^314 - 311*t^313 + 351*t^312 - 384*t^311 + 351*t^310 - 299*t^309 + 296*t^308 - 332*t^307 + 337*t^306 - 327*t^305 + 334*t^304 - 320*t^303 + 282*t^302 - 278*t^301 + 323*t^300 - 349*t^299 + 313*t^298 - 268*t^297 + 272*t^296 - 294*t^295 + 301*t^294 - 301*t^293 + 297*t^292 - 286*t^291 + 255*t^290 - 244*t^289 + 290*t^288 - 315*t^287 + 279*t^286 - 237*t^285 + 239*t^284 - 265*t^283 + 266*t^282 - 266*t^281 + 270*t^280 - 250*t^279 + 220*t^278 - 221*t^277 + 254*t^276 - 274*t^275 + 250*t^274 - 210*t^273 + 207*t^272 - 230*t^271 + 241*t^270 - 233*t^269 + 230*t^268 - 224*t^267 + 193*t^266 - 184*t^265 + 225*t^264 - 246*t^263 + 213*t^262 - 178*t^261 + 184*t^260 - 201*t^259 + 202*t^258 - 205*t^257 + 203*t^256 - 189*t^255 + 165*t^254 - 162*t^253 + 192*t^252 - 211*t^251 + 188*t^250 - 152*t^249 + 153*t^248 - 172*t^247 + 176*t^246 - 172*t^245 + 174*t^244 - 164*t^243 + 136*t^242 - 137*t^241 + 168*t^240 - 178*t^239 + 158*t^238 - 131*t^237 + 127*t^236 - 143*t^235 + 151*t^234 - 147*t^233 + 144*t^232 - 139*t^231 + 117*t^230 - 112*t^229 + 137*t^228 - 156*t^227 + 133*t^226 - 101*t^225 + 109*t^224 - 123*t^223 + 122*t^222 - 122*t^221 + 125*t^220 - 114*t^219 + 92*t^218 - 94*t^217 + 117*t^216 - 125*t^215 + 110*t^214 - 88*t^213 + 86*t^212 - 99*t^211 + 104*t^210 - 100*t^209 + 100*t^208 - 95*t^207 + 75*t^206 - 73*t^205 + 95*t^204 - 107*t^203 + 88*t^202 - 69*t^201 + 72*t^200 - 80*t^199 + 84*t^198 - 82*t^197 + 81*t^196 - 74*t^195 + 61*t^194 - 60*t^193 + 74*t^192 - 87*t^191 + 75*t^190 - 51*t^189 + 55*t^188 - 68*t^187 + 65*t^186 - 63*t^185 + 68*t^184 - 61*t^183 + 44*t^182 - 47*t^181 + 63*t^180 - 69*t^179 + 55*t^178 - 43*t^177 + 44*t^176 - 49*t^175 + 53*t^174 - 52*t^173 + 51*t^172 - 47*t^171 + 38*t^170 - 34*t^169 + 47*t^168 - 57*t^167 + 44*t^166 - 30*t^165 + 33*t^164 - 40*t^163 + 41*t^162 - 39*t^161 + 42*t^160 - 36*t^159 + 26*t^158 - 28*t^157 + 36*t^156 - 43*t^155 + 34*t^154 - 23*t^153 + 24*t^152 - 31*t^151 + 32*t^150 - 29*t^149 + 31*t^148 - 29*t^147 + 19*t^146 - 17*t^145 + 30*t^144 - 34*t^143 + 24*t^142 - 17*t^141 + 20*t^140 - 22*t^139 + 22*t^138 - 24*t^137 + 23*t^136 - 18*t^135 + 16*t^134 - 14*t^133 + 19*t^132 - 27*t^131 + 19*t^130 - 12*t^129 + 12*t^128 - 18*t^127 + 18*t^126 - 14*t^125 + 18*t^124 - 16*t^123 + 8*t^122 - 9*t^121 + 17*t^120 - 19*t^119 + 13*t^118 - 8*t^117 + 9*t^116 - 12*t^115 + 13*t^114 - 11*t^113 + 12*t^112 - 12*t^111 + 7*t^110 - 5*t^109 + 12*t^108 - 14*t^107 + 9*t^106 - 5*t^105 + 6*t^104 - 10*t^103 + 7*t^102 - 9*t^101 + 10*t^100 - 6*t^99 + 5*t^98 - 4*t^97 + 8*t^96 - 10*t^95 + 6*t^94 - 4*t^93 + 3*t^92 - 6*t^91 + 8*t^90 - 4*t^89 + 6*t^88 - 7*t^87 + 2*t^86 - t^85 + 6*t^84 - 8*t^83 + 4*t^82 - t^81 + 3*t^80 - 5*t^79 + 3*t^78 - 4*t^77 + 4*t^76 - 3*t^75 + 2*t^74 - t^73 + 4*t^72 - 5*t^71 + 3*t^70 - t^69 + t^68 - 4*t^67 + 2*t^66 - t^65 + 4*t^64 - 2*t^63 - t^61 + 3*t^60 - 3*t^59 + 2*t^58 + t^56 - 2*t^55 + t^54 - 2*t^53 + t^52 - 2*t^51 + t^50 + t^49 + 2*t^48 - 3*t^47 + t^44 - t^43 + t^42 + 2*t^40 - t^39 - t^38 - t^37 + t^36 - t^35 + t^34 + t^33 + t^32 - t^31 - t^29 - t^27 + t^25 + t^24 - t^23 - t^19 + t^16 - t^11 + t^9 + t^8 - t^3 - t^2 + 1)
NB Oscar can factorize the denominator in less than 0.5s, but the factorization is still more awkward to comprehend than the factorized form given by CoCoA.
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An obvious choice for the data-type would be the type which |
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On Wed, May 24, 2023 at 03:38:29AM -0700, JohnAAbbott wrote:
The hilbert series we compute are returned as a pair of polynomials (effectively _numerator_ and _denominator_, but we may want to allow common factors). The denominator is a product of factors of the form `(1-t^d)`; an easy generalization applies when we have a grading over `ZZ^k`.
Currently Oscar multiplies the denominator factors together producing a potentially large product (which may be expensive to factorize). In contrast, CoCoA leaves the denominator as a product of simple factors. _**Should Oscar also leave the denominator as a product of simple factors?**_
We have a generic type of FacElem which allows to have non-expanded
products (with large exponents)
… Here is a (contrived) concrete example: CoCoA produces the following result
```
(1 - t^1060) / ( (1-t^2)*(1-t^3)*(1-t^5)*(1-t^7)*(1-t^11)*(1-t^13)*(1-t^17)*(1-t^19)*(1-t^23)*(1-t^29)*(1-t^31)*(1-t^37)*(1-t^41)*(1-t^43)*(1-t^47)*(1-t^53)*(1-t^59)*(1-t^61)*(1-t^67)*(1-t^71)*(1-t^73)*(1-t^79)*(1-t^83)*(1-t^89)*(1-t^97) )
```
In contrast Oscar produces the equivalent result:
```
(-t^1060 + 1, -t^1060 + t^1058 + t^1057 - t^1052 - t^1051 + t^1049 - t^1044 + t^1041 + t^1037 - t^1036 - t^1035 + t^1033 + t^1031 + t^1029 - t^1028 - t^1027 - t^1026 + t^1025 - t^1024 + t^1023 + t^1022 + t^1021 - 2*t^1020 - t^1018 + t^1017 - t^1016 + 3*t^1013 - 2*t^1012 - t^1011 - t^1010 + 2*t^1009 - t^1008 + 2*t^1007 - t^1006 + 2*t^1005 - t^1004 - 2*t^1002 + 3*t^1001 - 3*t^1000 + t^999 + 2*t^997 - 4*t^996 + t^995 - 2*t^994 + 4*t^993 - t^992 + t^991 - 3*t^990 + 5*t^989 - 4*t^988 + t^987 - 2*t^986 + 3*t^985 - 4*t^984 + 4*t^983 - 3*t^982 + 5*t^981 - 3*t^980 + t^979 - 4*t^978 + 8*t^977 - 6*t^976 + t^975 - 2*t^974 + 7*t^973 - 6*t^972 + 4*t^971 - 8*t^970 + 6*t^969 - 3*t^968 + 4*t^967 - 6*t^966 + 10*t^965 - 8*t^964 + 4*t^963 - 5*t^962 + 6*t^961 - 10*t^960 + 9*t^959 - 7*t^958 + 10*t^957 - 6*t^956 + 5*t^955 - 9*t^954 + 14*t^953 - 12*t^952 + 5*t^951 - 7*t^950 + 12*t^949 - 12*t^948 + 11*t^947 - 13*t^946 + 12*t^945 - 9*t^944 + 8*t^943 - 13*t^942 + 19*t^941 - 17*t^940 + 9*t^939 - 8*t^938 + 16*t^937 - 18*t^936 + 14*t^935 - 18*t^934 + 18*t^933 - 12*t^932 + 12*t^931 - 19*t^930 + 27*t^929 - 19*t^928 + 14*t^927 - 16*t^926 + 18*t^925 - 23*t^924 + 24*t^923 - 22*t^922 + 22*t^921 - 20*t^920 + 17*t^919 - 24*t^918 + 34*t^917 - 30*t^916 + 17*t^915 - 19*t^914 + 29*t^913 - 31*t^912 + 29*t^911 - 32*t^910 + 31*t^909 - 24*t^908 + 23*t^907 - 34*t^906 + 43*t^905 - 36*t^904 + 28*t^903 - 26*t^902 + 36*t^901 - 42*t^900 + 39*t^899 - 41*t^898 + 40*t^897 - 33*t^896 + 30*t^895 - 44*t^894 + 57*t^893 - 47*t^892 + 34*t^891 - 38*t^890 + 47*t^889 - 51*t^888 + 52*t^887 - 53*t^886 + 49*t^885 - 44*t^884 + 43*t^883 - 55*t^882 + 69*t^881 - 63*t^880 + 47*t^879 - 44*t^878 + 61*t^877 - 68*t^876 + 63*t^875 - 65*t^874 + 68*t^873 - 55*t^872 + 51*t^871 - 75*t^870 + 87*t^869 - 74*t^868 + 60*t^867 - 61*t^866 + 74*t^865 - 81*t^864 + 82*t^863 - 84*t^862 + 80*t^861 - 72*t^860 + 69*t^859 - 88*t^858 + 107*t^857 - 95*t^856 + 73*t^855 - 75*t^854 + 95*t^853 - 100*t^852 + 100*t^851 - 104*t^850 + 99*t^849 - 86*t^848 + 88*t^847 - 110*t^846 + 125*t^845 - 117*t^844 + 94*t^843 - 92*t^842 + 114*t^841 - 125*t^840 + 122*t^839 - 122*t^838 + 123*t^837 - 109*t^836 + 101*t^835 - 133*t^834 + 156*t^833 - 137*t^832 + 112*t^831 - 117*t^830 + 139*t^829 - 144*t^828 + 147*t^827 - 151*t^826 + 143*t^825 - 127*t^824 + 131*t^823 - 158*t^822 + 178*t^821 - 168*t^820 + 137*t^819 - 136*t^818 + 164*t^817 - 174*t^816 + 172*t^815 - 176*t^814 + 172*t^813 - 153*t^812 + 152*t^811 - 188*t^810 + 211*t^809 - 192*t^808 + 162*t^807 - 165*t^806 + 189*t^805 - 203*t^804 + 205*t^803 - 202*t^802 + 201*t^801 - 184*t^800 + 178*t^799 - 213*t^798 + 246*t^797 - 225*t^796 + 184*t^795 - 193*t^794 + 224*t^793 - 230*t^792 + 233*t^791 - 241*t^790 + 230*t^789 - 207*t^788 + 210*t^787 - 250*t^786 + 274*t^785 - 254*t^784 + 221*t^783 - 220*t^782 + 250*t^781 - 270*t^780 + 266*t^779 - 266*t^778 + 265*t^777 - 239*t^776 + 237*t^775 - 279*t^774 + 315*t^773 - 290*t^772 + 244*t^771 - 255*t^770 + 286*t^769 - 297*t^768 + 301*t^767 - 301*t^766 + 294*t^765 - 272*t^764 + 268*t^763 - 313*t^762 + 349*t^761 - 323*t^760 + 278*t^759 - 282*t^758 + 320*t^757 - 334*t^756 + 327*t^755 - 337*t^754 + 332*t^753 - 296*t^752 + 299*t^751 - 351*t^750 + 384*t^749 - 351*t^748 + 311*t^747 - 316*t^746 + 344*t^745 - 368*t^744 + 367*t^743 - 364*t^742 + 360*t^741 - 335*t^740 + 328*t^739 - 375*t^738 + 419*t^737 - 389*t^736 + 333*t^735 - 342*t^734 + 385*t^733 - 396*t^732 + 391*t^731 - 401*t^730 + 393*t^729 - 356*t^728 + 357*t^727 - 409*t^726 + 447*t^725 - 415*t^724 + 364*t^723 - 369*t^722 + 411*t^721 - 427*t^720 + 418*t^719 - 426*t^718 + 422*t^717 - 384*t^716 + 377*t^715 - 437*t^714 + 478*t^713 - 437*t^712 + 389*t^711 - 397*t^710 + 432*t^709 - 450*t^708 + 445*t^707 - 450*t^706 + 440*t^705 - 406*t^704 + 406*t^703 - 453*t^702 + 498*t^701 - 468*t^700 + 405*t^699 - 410*t^698 + 461*t^697 - 470*t^696 + 456*t^695 - 469*t^694 + 466*t^693 - 423*t^692 + 414*t^691 - 478*t^690 + 518*t^689 - 475*t^688 + 421*t^687 - 430*t^686 + 471*t^685 - 482*t^684 + 474*t^683 - 482*t^682 + 474*t^681 - 438*t^680 + 428*t^679 - 482*t^678 + 531*t^677 - 487*t^676 + 427*t^675 - 439*t^674 + 482*t^673 - 487*t^672 + 479*t^671 - 491*t^670 + 480*t^669 - 441*t^668 + 434*t^667 - 486*t^666 + 529*t^665 - 493*t^664 + 430*t^663 - 436*t^662 + 486*t^661 - 494*t^660 + 472*t^659 - 484*t^658 + 487*t^657 - 440*t^656 + 422*t^655 - 485*t^654 + 531*t^653 - 481*t^652 + 422*t^651 - 438*t^650 + 480*t^649 - 475*t^648 + 468*t^647 - 479*t^646 + 470*t^645 - 433*t^644 + 416*t^643 - 470*t^642 + 515*t^641 - 470*t^640 + 410*t^639 - 422*t^638 + 464*t^637 - 465*t^636 + 444*t^635 - 458*t^634 + 459*t^633 - 414*t^632 + 396*t^631 - 450*t^630 + 493*t^629 - 448*t^628 + 388*t^627 - 399*t^626 + 443*t^625 - 440*t^624 + 418*t^623 - 427*t^622 + 437*t^621 - 393*t^620 + 363*t^619 - 421*t^618 + 466*t^617 - 414*t^616 + 353*t^615 - 375*t^614 + 416*t^613 - 400*t^612 + 380*t^611 - 401*t^610 + 400*t^609 - 354*t^608 + 334*t^607 - 384*t^606 + 420*t^605 - 377*t^604 + 322*t^603 - 333*t^602 + 375*t^601 - 364*t^600 + 336*t^599 - 350*t^598 + 363*t^597 - 317*t^596 + 287*t^595 - 337*t^594 + 376*t^593 - 329*t^592 + 274*t^591 - 292*t^590 + 329*t^589 - 315*t^588 + 283*t^587 - 300*t^586 + 316*t^585 - 269*t^584 + 236*t^583 - 285*t^582 + 322*t^581 - 276*t^580 + 219*t^579 - 242*t^578 + 283*t^577 - 251*t^576 + 223*t^575 - 249*t^574 + 260*t^573 - 213*t^572 + 185*t^571 - 227*t^570 + 257*t^569 - 214*t^568 + 168*t^567 - 185*t^566 + 218*t^565 - 197*t^564 + 160*t^563 - 181*t^562 + 201*t^561 - 159*t^560 + 123*t^559 - 159*t^558 + 194*t^557 - 153*t^556 + 100*t^555 - 126*t^554 + 162*t^553 - 126*t^552 + 91*t^551 - 116*t^550 + 140*t^549 - 96*t^548 + 58*t^547 - 93*t^546 + 125*t^545 - 82*t^544 + 36*t^543 - 64*t^542 + 98*t^541 - 57*t^540 + 17*t^539 - 51*t^538 + 73*t^537 - 29*t^536 - 5*t^535 - 26*t^534 + 49*t^533 - 12*t^532 - 26*t^531 + 26*t^529 + 12*t^528 - 49*t^527 + 26*t^526 + 5*t^525 + 29*t^524 - 73*t^523 + 51*t^522 - 17*t^521 + 57*t^520 - 98*t^519 + 64*t^518 - 36*t^517 + 82*t^516 - 125*t^515 + 93*t^514 - 58*t^513 + 96*t^512 - 140*t^511 + 116*t^510 - 91*t^509 + 126*t^508 - 162*t^507 + 126*t^506 - 100*t^505 + 153*t^504 - 194*t^503 + 159*t^502 - 123*t^501 + 159*t^500 - 201*t^499 + 181*t^498 - 160*t^497 + 197*t^496 - 218*t^495 + 185*t^494 - 168*t^493 + 214*t^492 - 257*t^491 + 227*t^490 - 185*t^489 + 213*t^488 - 260*t^487 + 249*t^486 - 223*t^485 + 251*t^484 - 283*t^483 + 242*t^482 - 219*t^481 + 276*t^480 - 322*t^479 + 285*t^478 - 236*t^477 + 269*t^476 - 316*t^475 + 300*t^474 - 283*t^473 + 315*t^472 - 329*t^471 + 292*t^470 - 274*t^469 + 329*t^468 - 376*t^467 + 337*t^466 - 287*t^465 + 317*t^464 - 363*t^463 + 350*t^462 - 336*t^461 + 364*t^460 - 375*t^459 + 333*t^458 - 322*t^457 + 377*t^456 - 420*t^455 + 384*t^454 - 334*t^453 + 354*t^452 - 400*t^451 + 401*t^450 - 380*t^449 + 400*t^448 - 416*t^447 + 375*t^446 - 353*t^445 + 414*t^444 - 466*t^443 + 421*t^442 - 363*t^441 + 393*t^440 - 437*t^439 + 427*t^438 - 418*t^437 + 440*t^436 - 443*t^435 + 399*t^434 - 388*t^433 + 448*t^432 - 493*t^431 + 450*t^430 - 396*t^429 + 414*t^428 - 459*t^427 + 458*t^426 - 444*t^425 + 465*t^424 - 464*t^423 + 422*t^422 - 410*t^421 + 470*t^420 - 515*t^419 + 470*t^418 - 416*t^417 + 433*t^416 - 470*t^415 + 479*t^414 - 468*t^413 + 475*t^412 - 480*t^411 + 438*t^410 - 422*t^409 + 481*t^408 - 531*t^407 + 485*t^406 - 422*t^405 + 440*t^404 - 487*t^403 + 484*t^402 - 472*t^401 + 494*t^400 - 486*t^399 + 436*t^398 - 430*t^397 + 493*t^396 - 529*t^395 + 486*t^394 - 434*t^393 + 441*t^392 - 480*t^391 + 491*t^390 - 479*t^389 + 487*t^388 - 482*t^387 + 439*t^386 - 427*t^385 + 487*t^384 - 531*t^383 + 482*t^382 - 428*t^381 + 438*t^380 - 474*t^379 + 482*t^378 - 474*t^377 + 482*t^376 - 471*t^375 + 430*t^374 - 421*t^373 + 475*t^372 - 518*t^371 + 478*t^370 - 414*t^369 + 423*t^368 - 466*t^367 + 469*t^366 - 456*t^365 + 470*t^364 - 461*t^363 + 410*t^362 - 405*t^361 + 468*t^360 - 498*t^359 + 453*t^358 - 406*t^357 + 406*t^356 - 440*t^355 + 450*t^354 - 445*t^353 + 450*t^352 - 432*t^351 + 397*t^350 - 389*t^349 + 437*t^348 - 478*t^347 + 437*t^346 - 377*t^345 + 384*t^344 - 422*t^343 + 426*t^342 - 418*t^341 + 427*t^340 - 411*t^339 + 369*t^338 - 364*t^337 + 415*t^336 - 447*t^335 + 409*t^334 - 357*t^333 + 356*t^332 - 393*t^331 + 401*t^330 - 391*t^329 + 396*t^328 - 385*t^327 + 342*t^326 - 333*t^325 + 389*t^324 - 419*t^323 + 375*t^322 - 328*t^321 + 335*t^320 - 360*t^319 + 364*t^318 - 367*t^317 + 368*t^316 - 344*t^315 + 316*t^314 - 311*t^313 + 351*t^312 - 384*t^311 + 351*t^310 - 299*t^309 + 296*t^308 - 332*t^307 + 337*t^306 - 327*t^305 + 334*t^304 - 320*t^303 + 282*t^302 - 278*t^301 + 323*t^300 - 349*t^299 + 313*t^298 - 268*t^297 + 272*t^296 - 294*t^295 + 301*t^294 - 301*t^293 + 297*t^292 - 286*t^291 + 255*t^290 - 244*t^289 + 290*t^288 - 315*t^287 + 279*t^286 - 237*t^285 + 239*t^284 - 265*t^283 + 266*t^282 - 266*t^281 + 270*t^280 - 250*t^279 + 220*t^278 - 221*t^277 + 254*t^276 - 274*t^275 + 250*t^274 - 210*t^273 + 207*t^272 - 230*t^271 + 241*t^270 - 233*t^269 + 230*t^268 - 224*t^267 + 193*t^266 - 184*t^265 + 225*t^264 - 246*t^263 + 213*t^262 - 178*t^261 + 184*t^260 - 201*t^259 + 202*t^258 - 205*t^257 + 203*t^256 - 189*t^255 + 165*t^254 - 162*t^253 + 192*t^252 - 211*t^251 + 188*t^250 - 152*t^249 + 153*t^248 - 172*t^247 + 176*t^246 - 172*t^245 + 174*t^244 - 164*t^243 + 136*t^242 - 137*t^241 + 168*t^240 - 178*t^239 + 158*t^238 - 131*t^237 + 127*t^236 - 143*t^235 + 151*t^234 - 147*t^233 + 144*t^232 - 139*t^231 + 117*t^230 - 112*t^229 + 137*t^228 - 156*t^227 + 133*t^226 - 101*t^225 + 109*t^224 - 123*t^223 + 122*t^222 - 122*t^221 + 125*t^220 - 114*t^219 + 92*t^218 - 94*t^217 + 117*t^216 - 125*t^215 + 110*t^214 - 88*t^213 + 86*t^212 - 99*t^211 + 104*t^210 - 100*t^209 + 100*t^208 - 95*t^207 + 75*t^206 - 73*t^205 + 95*t^204 - 107*t^203 + 88*t^202 - 69*t^201 + 72*t^200 - 80*t^199 + 84*t^198 - 82*t^197 + 81*t^196 - 74*t^195 + 61*t^194 - 60*t^193 + 74*t^192 - 87*t^191 + 75*t^190 - 51*t^189 + 55*t^188 - 68*t^187 + 65*t^186 - 63*t^185 + 68*t^184 - 61*t^183 + 44*t^182 - 47*t^181 + 63*t^180 - 69*t^179 + 55*t^178 - 43*t^177 + 44*t^176 - 49*t^175 + 53*t^174 - 52*t^173 + 51*t^172 - 47*t^171 + 38*t^170 - 34*t^169 + 47*t^168 - 57*t^167 + 44*t^166 - 30*t^165 + 33*t^164 - 40*t^163 + 41*t^162 - 39*t^161 + 42*t^160 - 36*t^159 + 26*t^158 - 28*t^157 + 36*t^156 - 43*t^155 + 34*t^154 - 23*t^153 + 24*t^152 - 31*t^151 + 32*t^150 - 29*t^149 + 31*t^148 - 29*t^147 + 19*t^146 - 17*t^145 + 30*t^144 - 34*t^143 + 24*t^142 - 17*t^141 + 20*t^140 - 22*t^139 + 22*t^138 - 24*t^137 + 23*t^136 - 18*t^135 + 16*t^134 - 14*t^133 + 19*t^132 - 27*t^131 + 19*t^130 - 12*t^129 + 12*t^128 - 18*t^127 + 18*t^126 - 14*t^125 + 18*t^124 - 16*t^123 + 8*t^122 - 9*t^121 + 17*t^120 - 19*t^119 + 13*t^118 - 8*t^117 + 9*t^116 - 12*t^115 + 13*t^114 - 11*t^113 + 12*t^112 - 12*t^111 + 7*t^110 - 5*t^109 + 12*t^108 - 14*t^107 + 9*t^106 - 5*t^105 + 6*t^104 - 10*t^103 + 7*t^102 - 9*t^101 + 10*t^100 - 6*t^99 + 5*t^98 - 4*t^97 + 8*t^96 - 10*t^95 + 6*t^94 - 4*t^93 + 3*t^92 - 6*t^91 + 8*t^90 - 4*t^89 + 6*t^88 - 7*t^87 + 2*t^86 - t^85 + 6*t^84 - 8*t^83 + 4*t^82 - t^81 + 3*t^80 - 5*t^79 + 3*t^78 - 4*t^77 + 4*t^76 - 3*t^75 + 2*t^74 - t^73 + 4*t^72 - 5*t^71 + 3*t^70 - t^69 + t^68 - 4*t^67 + 2*t^66 - t^65 + 4*t^64 - 2*t^63 - t^61 + 3*t^60 - 3*t^59 + 2*t^58 + t^56 - 2*t^55 + t^54 - 2*t^53 + t^52 - 2*t^51 + t^50 + t^49 + 2*t^48 - 3*t^47 + t^44 - t^43 + t^42 + 2*t^40 - t^39 - t^38 - t^37 + t^36 - t^35 + t^34 + t^33 + t^32 - t^31 - t^29 - t^27 + t^25 + t^24 - t^23 - t^19 + t^16 - t^11 + t^9 + t^8 - t^3 - t^2 + 1)
```
NB Oscar can factorize the denominator in less than 0.5s, but the factorization is still more awkward to comprehend than the factorized form given by CoCoA.
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Putative summary: we are happy to have the denominator in factorized form (probably as a FacElem) Oh! Now I wonder what to do for the reduced form. Perhaps we are interested only in the reduced numerator? |
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On Wed, May 31, 2023 at 01:31:17AM -0700, JohnAAbbott wrote:
**Putative summary:** we are happy to have the denominator in factorized form (probably as a _FacElem_)
Oh! Now I wonder what to do for the reduced form. Perhaps we are interested only in the _reduced numerator_?
there simplify for FacElem's as well - at least it can be implemented
for some types
Actually, there is a rich interface for them
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On Wed, May 31, 2023 at 01:31:17AM -0700, JohnAAbbott wrote:
**Putative summary:** we are happy to have the denominator in factorized form (probably as a _FacElem_)
Oh! Now I wonder what to do for the reduced form. Perhaps we are interested only in the _reduced numerator_?
julia> FacElem(Dict(x=>100, y=>10))
Factored element with data
Dict{AbstractAlgebra.Generic.MPoly{NfAbsNSElem}, ZZRingElem}(x => 100, y
=> 10)
julia> evaluate(ans)
x^100*y^10
julia> FacElem(Dict(x=>100, y=>-10))
Factored element with data
Dict{AbstractAlgebra.Generic.MPoly{NfAbsNSElem}, ZZRingElem}(x => 100, y
=> -10)
julia> evaluate(ans)
ERROR: DomainError with -5:
exponent must be >= 0
...
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One gets a more entertaining error with the following input: |
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On Wed, May 31, 2023 at 06:52:15AM -0700, JohnAAbbott wrote:
One gets a more entertaining error with the following input:
```
julia> P,(x,y) = LaurentPolynomialRing(QQ,["x","y"]);
julia> FacElem(Dict(x=>100, y=>-10))
julia> evaluate(ans)
ERROR: MethodError: no method matching copy(::AbstractAlgebra.Generic.LaurentMPolyWrap{QQFieldElem, QQMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{QQFieldElem, QQMPolyRing}})
...
Laurent polynomials are incomplete due to not having been used so
much...
… ```
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After speaking to Bigatti and Robbiano, the proposal for the output value is a pair: non-reduced numerator, and denominator as a factorization. Note that the factors of the denominator all have a special form: namely, |
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On Wed, May 31, 2023 at 06:52:15AM -0700, JohnAAbbott wrote:
One gets a more entertaining error with the following input:
```
julia> P,(x,y) = LaurentPolynomialRing(QQ,["x","y"]);
julia> FacElem(Dict(x=>100, y=>-10))
julia> evaluate(ans)
ERROR: MethodError: no method matching copy(::AbstractAlgebra.Generic.LaurentMPolyWrap{QQFieldElem, QQMPolyRingElem, AbstractAlgebra.Generic.LaurentMPolyWrapRing{QQFieldElem, QQMPolyRing}})
...
```
```
Nemo.copy(a::AbstractAlgebra.Generic.LaurentMPolyWrap) = a
```
solve this
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The hilbert series we compute are returned as a pair of polynomials (effectively numerator and denominator, but we may want to allow common factors). The denominator is a product of factors of the form
(1-t^d); an easy generalization applies when we have a grading overZZ^k.Currently Oscar multiplies the denominator factors together producing a potentially large product (which may be expensive to factorize). In contrast, CoCoA leaves the denominator as a product of simple factors. Should Oscar also leave the denominator as a product of simple factors?
Here is a (contrived) concrete example: CoCoA produces the following result
In contrast Oscar produces the equivalent result:
NB Oscar can factorize the denominator in less than 0.5s, but the factorization is still more awkward to comprehend than the factorized form given by CoCoA.
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