¹à½ç½ø¤ÎÀßÄê

¹à¤ÏÆâÉô¤Ç¤Ï, ³ÆÊÑ¿ô¤Ë´Ø¤¹¤ë»Ø¿ô¤òÀ®Ê¬¤È¤¹¤ëÀ°¿ô¥Ù¥¯¥È¥ë¤È¤·¤Æɽ¸½¤µ¤ì ¤ë. ¿¹à¼°¤òʬ»¶É½¸½Â¿¹à¼°¤ËÊÑ´¹¤¹¤ëºÝ, ³ÆÊÑ¿ô¤¬¤É¤ÎÀ®Ê¬¤ËÂбþ¤¹¤ë¤«¤ò »ØÄꤹ¤ë¤Î¤¬, ÊÑ¿ô¥ê¥¹¥È¤Ç¤¢¤ë. ¤µ¤é¤Ë, ¤½¤ì¤éÀ°¿ô¥Ù¥¯¥È¥ë¤ÎÁ´½ç½ø¤ò »ØÄꤹ¤ë¤Î¤¬¹à½ç½ø¤Î·¿¤Ç¤¢¤ë. ¹à½ç½ø·¿¤Ï, ¿ô, ¿ô¤Î¥ê¥¹¥È¤¢¤ë¤¤¤Ï ¹ÔÎó¤Çɽ¸½¤µ¤ì¤ë.

´ðËÜŪ¤Ê¹à½ç½ø·¿¤È¤·¤Æ¼¡¤Î 3 ¤Ä¤¬¤¢¤ë.

0 (DegRevLex; Á´¼¡¿ôµÕ¼­½ñ¼°½ç½ø)

°ìÈ̤Ë, ¤³¤Î½ç½ø¤Ë¤è¤ë¥°¥ì¥Ö¥Ê´ðÄì·×»»¤¬ºÇ¤â¹â®¤Ç¤¢¤ë. ¤¿¤À¤·, ÊýÄø¼°¤ò²ò¤¯¤È¤¤¤¦ÌÜŪ¤ËÍѤ¤¤ë¤³¤È¤Ï, °ìÈ̤ˤϤǤ­¤Ê¤¤. ¤³¤Î ½ç½ø¤Ë¤è¤ë¥°¥ì¥Ö¥Ê´ðÄì¤Ï, ²ò¤Î¸Ä¿ô¤Î·×»», ¥¤¥Ç¥¢¥ë¤Î¥á¥ó¥Ð¥·¥Ã¥×¤ä, ¾¤ÎÊÑ¿ô½ç½ø¤Ø¤Î´ðÄìÊÑ´¹¤Î¤¿¤á¤Î¥½¡¼¥¹¤È¤·¤ÆÍѤ¤¤é¤ì¤ë.

1 (DegLex; Á´¼¡¿ô¼­½ñ¼°½ç½ø)

¤³¤Î½ç½ø¤â, ¼­½ñ¼°½ç½ø¤ËÈæ¤Ù¤Æ¹â®¤Ë¥°¥ì¥Ö¥Ê´ðÄì¤òµá¤á¤ë¤³¤È¤¬¤Ç¤­¤ë¤¬, DegRevLex ¤ÈƱÍÍľÀܤ½¤Î·ë²Ì¤òÍѤ¤¤ë¤³¤È¤Ïº¤Æñ¤Ç¤¢¤ë. ¤·¤«¤·, ¼­½ñ¼°½ç½ø¤Î¥°¥ì¥Ö¥Ê´ðÄì¤òµá¤á¤ëºÝ¤Ë, ÀƼ¡²½¸å¤Ë¤³¤Î½ç½ø¤Ç¥°¥ì¥Ö¥Ê´ðÄì ¤òµá¤á¤Æ¤¤¤ë.

2 (Lex; ¼­½ñ¼°½ç½ø)

¤³¤Î½ç½ø¤Ë¤è¤ë¥°¥ì¥Ö¥Ê´ðÄì¤Ï, ÊýÄø¼°¤ò²ò¤¯¾ì¹ç¤ËºÇŬ¤Î·Á¤Î´ðÄì¤òÍ¿¤¨¤ë¤¬ ·×»»»þ´Ö¤¬¤«¤«¤ê²á¤®¤ë¤Î¤¬ÆñÅÀ¤Ç¤¢¤ë. ÆäË, ²ò¤¬Í­¸Â¸Ä¤Î¾ì¹ç, ·ë²Ì¤Î ·¸¿ô¤¬¶Ë¤á¤ÆĹÂç¤Ê¿ÇÜĹ¿ô¤Ë¤Ê¤ë¾ì¹ç¤¬Â¿¤¤. ¤³¤Î¾ì¹ç, gr(), hgr() ¤Ë¤è¤ë·×»»¤¬¶Ë¤á¤ÆÍ­¸ú¤Ë¤Ê¤ë¾ì¹ç¤¬Â¿¤¤.

¤³¤ì¤é¤òÁȤ߹ç¤ï¤»¤Æ¥ê¥¹¥È¤Ç»ØÄꤹ¤ë¤³¤È¤Ë¤è¤ê, ÍÍ¡¹¤Ê¾Ãµî½ç½ø¤¬»ØÄê¤Ç¤­¤ë. ¤³¤ì¤Ï,

[[O1,L1],[O2,L2],...]

¤Ç»ØÄꤵ¤ì¤ë. Oi ¤Ï 0, 1, 2 ¤Î¤¤¤º¤ì¤«¤Ç, Li ¤ÏÊÑ¿ô¤Î¸Ä ¿ô¤òɽ¤¹. ¤³¤Î»ØÄê¤Ï, ÊÑ¿ô¤òÀèƬ¤«¤é L1, L2 , ...¸Ä ¤º¤Ä¤ÎÁȤËʬ¤±, ¤½¤ì¤¾¤ì¤ÎÊÑ¿ô¤Ë´Ø¤·, ½ç¤Ë O1, O2, ...¤Î¹à½ç½ø·¿¤ÇÂç¾®¤¬·èÄꤹ¤ë¤Þ¤ÇÈæ³Ó¤¹¤ë¤³¤È¤ò°ÕÌ£¤¹¤ë. ¤³¤Î·¿¤Î ½ç½ø¤Ï°ìÈ̤˾õî½ç½ø¤È¸Æ¤Ð¤ì¤ë.

¤µ¤é¤Ë, ¹ÔÎó¤Ë¤è¤ê¹à½ç½ø¤ò»ØÄꤹ¤ë¤³¤È¤¬¤Ç¤­¤ë. °ìÈ̤Ë, n ¹Ô m Îó¤Î¼Â¿ô¹ÔÎó M ¤¬¼¡¤ÎÀ­¼Á¤ò»ý¤Ä¤È¤¹¤ë.

1. Ťµ m ¤ÎÀ°¿ô¥Ù¥¯¥È¥ë v ¤ËÂФ· Mv=0 ¤È v=0 ¤ÏƱÃÍ.
2. ÈóÉéÀ®Ê¬¤ò»ý¤ÄŤµ m ¤Î 0 ¤Ç¤Ê¤¤À°¿ô¥Ù¥¯¥È¥ë v ¤ËÂФ·, Mv ¤Î 0 ¤Ç¤Ê¤¤ºÇ½é¤ÎÀ®Ê¬¤ÏÈóÉé.

¤³¤Î»þ, 2 ¤Ä¤Î¥Ù¥¯¥È¥ë t, s ¤ËÂФ·, t>s ¤ò, M(t-s) ¤Î 0 ¤Ç¤Ê¤¤ºÇ½é¤ÎÀ®Ê¬¤¬ÈóÉé, ¤ÇÄêµÁ¤¹¤ë¤³¤È¤Ë¤è¤ê¹à½ç½ø¤¬ÄêµÁ¤Ç¤­¤ë.

¹à½ç½ø·¿¤Ï, gr() ¤Ê¤É¤Î°ú¿ô¤È¤·¤Æ»ØÄꤵ¤ì¤ë¾, ÁȤ߹þ¤ßÈ¡¿ô dp_ord() ¤Ç»ØÄꤵ¤ì, ¤µ¤Þ¤¶¤Þ¤ÊÈ¡¿ô¤Î¼Â¹Ô¤ÎºÝ¤Ë»²¾È¤µ¤ì¤ë.

¤³¤ì¤é¤Î½ç½ø¤Î¶ñÂÎŪ¤ÊÄêµÁ¤ª¤è¤Ó¥°¥ì¥Ö¥Ê´ðÄì¤Ë´Ø¤¹¤ë¹¹¤Ë¾Ü¤·¤¤²òÀâ¤Ï [Becker,Weispfenning] ¤Ê¤É¤ò»²¾È¤Î¤³¤È.

¹à½ç½ø·¿¤ÎÀßÄê¤Î¾¤Ë, ÊÑ¿ô¤Î½ç½ø¼«ÂΤâ·×»»»þ´Ö¤ËÂ礭¤Ê±Æ¶Á¤òÍ¿¤¨¤ë.

[90] B=[x^10-t,x^8-z,x^31-x^6-x-y]$
[91] gr(B,[x,y,z,t],2);
[x^2-2*y^7+(-41*t^2-13*t-1)*y^2+(2*t^17-12*t^14+42*t^12+30*t^11-168*t^9
-40*t^8+70*t^7+252*t^6+30*t^5-140*t^4-168*t^3+2*t^2-12*t+16)*z^2*y
+(-12*t^16+72*t^13-28*t^11-180*t^10+112*t^8+240*t^7+28*t^6-127*t^5
-167*t^4-55*t^3+30*t^2+58*t-15)*z^4,
(y+t^2*z^2)*x+y^7+(20*t^2+6*t+1)*y^2+(-t^17+6*t^14-21*t^12-15*t^11+84*t^9
+20*t^8-35*t^7-126*t^6-15*t^5+70*t^4+84*t^3-t^2+5*t-9)*z^2*y+(6*t^16-36*t^13
+14*t^11+90*t^10-56*t^8-120*t^7-14*t^6+64*t^5+84*t^4+27*t^3-16*t^2-30*t+7)*z^4,
(t^3-1)*x-y^6+(-6*t^13+24*t^10-20*t^8-36*t^7+40*t^5+24*t^4-6*t^3-20*t^2-6*t-1)*y
+(t^17-6*t^14+9*t^12+15*t^11-36*t^9-20*t^8-5*t^7+54*t^6+15*t^5+10*t^4-36*t^3
-11*t^2-5*t+9)*z^2,
-y^8-8*t*y^3+16*z^2*y^2+(-8*t^16+48*t^13-56*t^11-120*t^10+224*t^8+160*t^7
-56*t^6-336*t^5-112*t^4+112*t^3+224*t^2+24*t-56)*z^4*y+(t^24-8*t^21+20*t^19
+28*t^18-120*t^16-56*t^15+14*t^14+300*t^13+70*t^12-56*t^11-400*t^10-84*t^9
+84*t^8+268*t^7+84*t^6-56*t^5-63*t^4-36*t^3+46*t^2-12*t+1)*z,
2*t*y^5+z*y^2+(-2*t^11+8*t^8-20*t^6-12*t^5+40*t^3+8*t^2-10*t-20)*z^3*y+8*t^14
-32*t^11+48*t^8-t^7-32*t^5-6*t^4+9*t^2-t,
-z*y^3+(t^7-2*t^4+3*t^2+t)*y+(-2*t^6+4*t^3+2*t-2)*z^2,
2*t^2*y^3+z^2*y^2+(-2*t^5+4*t^2-6)*z^4*y+(4*t^8-t^7-8*t^5+2*t^4-4*t^3+5*t^2-t)*z,
z^3*y^2+2*t^3*y+(-t^7+2*t^4+t^2-t)*z^2,
-t*z*y^2-2*z^3*y+t^8-2*t^5-t^3+t^2,
-t^3*y^2-2*t^2*z^2*y+(t^6-2*t^3-t+1)*z^4,
z^5-t^4]
[93] gr(B,[t,z,y,x],2);
[x^10-t,x^8-z,x^31-x^6-x-y]

ÊÑ¿ô½ç½ø [x,y,z,t] ¤Ë¤ª¤±¤ë¥°¥ì¥Ö¥Ê´ðÄì¤Ï, ´ðÄì¤Î¿ô¤â¿¤¯, ¤½¤ì¤¾¤ì¤Î ¼°¤âÂ礭¤¤. ¤·¤«¤·, ½ç½ø [t,z,y,x] ¤Ë¤â¤È¤Ç¤Ï, B ¤¬¤¹¤Ç¤Ë ¥°¥ì¥Ö¥Ê´ðÄì¤È¤Ê¤Ã¤Æ¤¤¤ë. Â绨ÇĤˤ¤¤¨¤Ð, ¼­½ñ¼°½ç½ø¤Ç¥°¥ì¥Ö¥Ê´ðÄì¤òµá¤á¤ë ¤³¤È¤Ï, º¸Â¦¤Î (½ç½ø¤Î¹â¤¤) ÊÑ¿ô¤ò, ±¦Â¦¤Î (½ç½ø¤ÎÄ㤤) ÊÑ¿ô¤Ç½ñ¤­É½¤¹ ¤³¤È¤Ç¤¢¤ê, ¤³¤ÎÎã¤Î¾ì¹ç¤Ï, t, z, y ¤¬´û¤Ë x ¤Çɽ¤µ¤ì¤Æ¤¤¤ë¤³¤È¤«¤é¤³¤Î¤è¤¦¤Ê¶Ëü¤Ê·ë²Ì¤È¤Ê¤Ã¤¿¤ï¤±¤Ç¤¢¤ë. ¼ÂºÝ¤Ë¸½¤ì¤ë·×»»¤Ë¤ª¤¤¤Æ¤Ï, ¤³¤Î¤è¤¦¤ËÁª¤Ö¤Ù¤­ÊÑ¿ô½ç½ø¤¬ÌÀ¤é¤«¤Ç¤¢¤ë ¤³¤È¤Ï¾¯¤Ê¤¯, »î¹Ôºø¸í¤¬É¬Íפʾì¹ç¤â¤¢¤ë.

Go to the first, previous, next, last section, table of contents.