代数方程的虚根判定问题

dingjifen

判定下列方程有无虚根：

（1）1+x/2!+x²/4!+x³/6!+……+xⁿ/(2n)!=0

（2）1+x/3!+x²/5!+x³/7!+……+xⁿ/(2n+1)!=0

补充内容 (2020-1-20 12:47):
第一个方程n=3时有虚根，第二个方程n=2时有虚根。n=其它自然数呢？

补充内容 (2020-1-20 12:56):
第二个方程n=3时有虚根。

补充内容 (2020-1-20 21:25):
可以证明——当方程次数n相当大时，以上两个方程无虚根。

dingjifen

通过验证，初步判定以上两个方程无虚根，但无法证明。

补充内容 (2020-1-20 12:45):
第二个方程n=2时有虚根。


补充内容 (2020-1-20 13:02):
第二个方程n=3时有虚根。

kastin

详见杨翠红、朱思铭、梁肇军的论文《多项式代数方程根的完全分类及其应用》。

dingjifen

kastin 发表于 2020-1-19 23:03
详见杨翠红、朱思铭、梁肇军的论文《多项式代数方程根的完全分类及其应用》。

该论文只是给出了五、六次方程根的判定，而对七次及上方程根是提供软件包来判定，这算不上是证明。

mathe

f=1+x/2!+x^2/4!+x^3/6!
polroots(f)
%11 = [-2.4646042998750414383960323257279253689 + 0.E-38*I, -13.767697850062479280801983837136037316 - 10.128506369060360412786701594442666541*I, -13.767697850062479280801983837136037316 + 10.128506369060360412786701594442666541*I]~

dingjifen

mathe 发表于 2020-1-20 08:10
f=1+x/2!+x^2/4!+x^3/6!
polroots(f)
%11 = [-2.4646042998750414383960323257279253689 + 0.E-38*I, -13 ...

1、看来第一个方程n=3时有虚根，第二个方程n=2、3时有虚根。

2、管理员的计算软件能计算定积分否？

dingjifen

kastin 发表于 2020-1-19 23:03
详见杨翠红、朱思铭、梁肇军的论文《多项式代数方程根的完全分类及其应用》。

此论文对判定七次及上方程根的计算太复杂了。此论文只能判定n=具体整数时，多项式代数方程根的完全分类。对于n=任意整数时，无法判定多项式代数方程根的完全分类。所以说此论文判定不算证明。

zeroieme

1. Table[{"n=" <> ToString[n], Solve[(\!\(
   
2. \*UnderoverscriptBox[\(\[Sum]\), \(i = 0\), \(n\)]
   
3. \*FractionBox[
   
4. SuperscriptBox[\(x\), \(i\)], \(\((2  i)\)!\)]\)) == 0] // N}, {n, 20}]
   
5. Table[{"n=" <> ToString[n], Solve[(\!\(
   
6. \*UnderoverscriptBox[\(\[Sum]\), \(i = 0\), \(n\)]
   
7. \*FractionBox[
   
8. SuperscriptBox[\(x\), \(i\)], \(\((2  i + 1)\)!\)]\)) == 0] // N}, {n,
   
9.    20}]

复制代码

{{n=1,{{x->-2.}}},{n=2,{{x->-9.4641},{x->-2.5359}}},{n=3,{{x->-2.4646},{x->-13.7677+10.1285 I},{x->-13.7677-10.1285 I}}},{n=4,{{x->-17.7741-24.345 I},{x->-17.7741+24.345 I},{x->-17.9844},{x->-2.46748}}},{n=5,{{x->-2.4674},{x->-24.9706-3.62429 I},{x->-24.9706+3.62429 I},{x->-18.7957-44.2348 I},{x->-18.7957+44.2348 I}}},{n=6,{{x->-21.9634},{x->-2.4674},{x->-37.7285-18.4539 I},{x->-37.7285+18.4539 I},{x->-16.056-68.9415 I},{x->-16.056+68.9415 I}}},{n=7,{{x->-46.266},{x->-22.2317},{x->-2.4674},{x->-46.3818-37.2016 I},{x->-46.3818+37.2016 I},{x->-9.13567-98.1489 I},{x->-9.13567+98.1489 I}}},{n=8,{{x->-22.2048},{x->-2.4674},{x->-56.4911-12.8299 I},{x->-56.4911+12.8299 I},{x->-53.4886-60.9096 I},{x->-53.4886+60.9096 I},{x->2.31588 -131.577 I},{x->2.31588 +131.577 I}}},{n=9,{{x->-58.8535},{x->-22.2067},{x->-2.4674},{x->-71.6605-29.9017 I},{x->-71.6605+29.9017 I},{x->-58.1434-89.5125 I},{x->-58.1434+89.5125 I},{x->18.5677 -168.99 I},{x->18.5677 +168.99 I}}},{n=10,{{x->-82.6141},{x->-62.3856},{x->-22.2066},{x->-2.4674},{x->-85.0482-53.3923 I},{x->-85.0482+53.3923 I},{x->-59.9508-122.778 I},{x->-59.9508+122.778 I},{x->39.8358 -210.184 I},{x->39.8358 +210.184 I}}},{n=11,{{x->-61.6159},{x->-22.2066},{x->-2.4674},{x->-98.6916-22.5118 I},{x->-98.6916+22.5118 I},{x->-96.8824-81.4904 I},{x->-96.8824+81.4904 I},{x->-58.5797-160.545 I},{x->-58.5797+160.545 I},{x->66.2986 -254.984 I},{x->66.2986 +254.984 I}}},{n=12,{{x->-107.803},{x->-61.6918},{x->-22.2066},{x->-2.4674},{x->-116.395-44.8501 I},{x->-116.395+44.8501 I},{x->-106.879-114.381 I},{x->-106.879+114.381 I},{x->-53.7487-202.661 I},{x->-53.7487+202.661 I},{x->98.1066 -303.239 I},{x->98.1066 +303.239 I}}},{n=13,{{x->-61.6845},{x->-22.2066},{x->-2.4674},{x->-134.078-72.6475 I},{x->-134.078+72.6475 I},{x->-123.258-11.8505 I},{x->-123.258+11.8505 I},{x->-114.654-151.948 I},{x->-114.654+151.948 I},{x->-45.2189-248.984 I},{x->-45.2189+248.984 I},{x->135.388 -354.813 I},{x->135.388 +354.813 I}}},{n=14,{{x->-119.795},{x->-61.6851},{x->-22.2066},{x->-2.4674},{x->-150.337-105.303 I},{x->-150.337+105.303 I},{x->-150.14-34.4196 I},{x->-150.14+34.4196 I},{x->-119.918-194.081 I},{x->-119.918+194.081 I},{x->-32.7842-299.385 I},{x->-32.7842+299.385 I},{x->178.256 -409.586 I},{x->178.256 +409.586 I}}},{n=15,{{x->-164.65},{x->-121.063},{x->-61.685},{x->-22.2066},{x->-2.4674},{x->-172.058-62.5943 I},{x->-172.058+62.5943 I},{x->-165.032-142.654 I},{x->-165.032+142.654 I},{x->-122.418-240.675 I},{x->-122.418+240.675 I},{x->-16.2646-353.746 I},{x->-16.2646+353.746 I},{x->226.809 -467.449 I},{x->226.809 +467.449 I}}},{n=16,{{x->-120.886},{x->-61.685},{x->-22.2066},{x->-2.4674},{x->-193.618-94.9724 I},{x->-193.618+94.9724 I},{x->-184.587-24.6972 I},{x->-184.587+24.6972 I},{x->-177.876-184.66 I},{x->-177.876+184.66 I},{x->-121.929-291.631 I},{x->-121.929+291.631 I},{x->4.49902 -411.955 I},{x->4.49902 +411.955 I},{x->281.133 -528.304 I},{x->281.133 +528.304 I}}},{n=17,{{x->-191.679},{x->-120.904},{x->-61.685},{x->-22.2066},{x->-2.4674},{x->-214.126-132.192 I},{x->-214.126+132.192 I},{x->-211.495-50.3825 I},{x->-211.495+50.3825 I},{x->-188.612-231.244 I},{x->-188.612+231.244 I},{x->-118.252-346.852 I},{x->-118.252+346.852 I},{x->29.6481 -473.911 I},{x->29.6481 +473.911 I},{x->341.308 -592.062 I},{x->341.308 +592.062 I}}},{n=18,{{x->-223.592},{x->-203.3},{x->-120.903},{x->-61.685},{x->-22.2066},{x->-2.4674},{x->-238.171-83.1045 I},{x->-238.171+83.1045 I},{x->-233.244-174.09 I},{x->-233.244+174.09 I},{x->-197.013-282.33 I},{x->-197.013+282.33 I},{x->-111.21-406.247 I},{x->-111.21+406.247 I},{x->59.3098 -539.519 I},{x->59.3098 +539.519 I},{x->407.404 -658.641 I},{x->407.404 +658.641 I}}},{n=19,{{x->-199.549},{x->-120.903},{x->-61.685},{x->-22.2066},{x->-2.4674},{x->-263.725-120.283 I},{x->-263.725+120.283 I},{x->-254.692-37.407 I},{x->-254.692+37.407 I},{x->-250.745-220.613 I},{x->-250.745+220.613 I},{x->-202.876-337.843 I},{x->-202.876+337.843 I},{x->-100.642-469.729 I},{x->-100.642+469.729 I},{x->93.599 -608.691 I},{x->93.599 +608.691 I},{x->479.486 -727.965 I},{x->479.486 +727.965 I}}},{n=20,{{x->-270.49},{x->-199.897},{x->-120.903},{x->-61.685},{x->-22.2066},{x->-2.4674},{x->-288.348-162.11 I},{x->-288.348+162.11 I},{x->-284.233-69.6935 I},{x->-284.233+69.6935 I},{x->-266.408-271.71 I},{x->-266.408+271.71 I},{x->-206.016-397.71 I},{x->-206.016+397.71 I},{x->-86.4031-537.217 I},{x->-86.4031+537.217 I},{x->132.62 -681.346 I},{x->132.62 +681.346 I},{x->557.612 -799.966 I},{x->557.612 +799.966 I}}}}

{{n=1,{{x->-6.}}},{n=2,{{x->-10.-4.47214 I},{x->-10.+4.47214 I}}},{n=3,{{x->-9.47804},{x->-16.261+16.3504 I},{x->-16.261-16.3504 I}}},{n=4,{{x->-18.7264-33.693 I},{x->-18.7264+33.693 I},{x->-24.6329},{x->-9.91425}}},{n=5,{{x->-9.86681},{x->-32.1468-11.6869 I},{x->-32.1468+11.6869 I},{x->-17.9198-56.0056 I},{x->-17.9198+56.0056 I}}},{n=6,{{x->-35.7407},{x->-9.86974},{x->-42.0519-27.0435 I},{x->-42.0519+27.0435 I},{x->-13.1428-83.0009 I},{x->-13.1428+83.0009 I}}},{n=7,{{x->-50.4912},{x->-41.1657},{x->-9.8696},{x->-50.2421-48.422 I},{x->-50.2421+48.422 I},{x->-3.9946-114.351 I},{x->-3.9946+114.351 I}}},{n=8,{{x->-39.3448},{x->-9.8696},{x->-65.0718-20.6442 I},{x->-65.0718+20.6442 I},{x->-56.1482-74.6212 I},{x->-56.1482+74.6212 I},{x->9.82721 -149.799 I},{x->9.82721 +149.799 I}}},{n=9,{{x->-74.0813},{x->-39.4916},{x->-9.8696},{x->-78.417-41.1934 I},{x->-78.417+41.1934 I},{x->-59.4245-105.572 I},{x->-59.4245+105.572 I},{x->28.5628 -189.125 I},{x->28.5628 +189.125 I}}},{n=10,{{x->-39.4774},{x->-9.8696},{x->-91.1618-66.8126 I},{x->-91.1618+66.8126 I},{x->-86.8913-13.0751 I},{x->-86.8913+13.0751 I},{x->-59.6813-141.109 I},{x->-59.6813+141.109 I},{x->52.4079 -232.143 I},{x->52.4079 +232.143 I}}},{n=11,{{x->-86.9529},{x->-39.4785},{x->-9.8696},{x->-107.621-32.3728 I},{x->-107.621+32.3728 I},{x->-102.142-97.3441 I},{x->-102.142+97.3441 I},{x->-56.6125-181.068 I},{x->-56.6125+181.068 I},{x->81.526 -278.689 I},{x->81.526 +278.689 I}}},{n=12,{{x->-120.597},{x->-89.1576},{x->-39.4784},{x->-9.8696},{x->-125.482-58.2035 I},{x->-125.482+58.2035 I},{x->-111.063-132.588 I},{x->-111.063+132.588 I},{x->-49.9597-225.305 I},{x->-49.9597+225.305 I},{x->116.056 -328.619 I},{x->116.056 +328.619 I}}},{n=13,{{x->-88.7918},{x->-39.4784},{x->-9.8696},{x->-142.368-88.3917 I},{x->-142.368+88.3917 I},{x->-138.562-23.911 I},{x->-138.562+23.911 I},{x->-117.616-172.451 I},{x->-117.616+172.451 I},{x->-39.5015-273.683 I},{x->-39.5015+273.683 I},{x->156.118 -381.807 I},{x->156.118 +381.807 I}}},{n=14,{{x->-147.122},{x->-88.8298},{x->-39.4784},{x->-9.8696},{x->-160.694-47.9411 I},{x->-160.694+47.9411 I},{x->-157.899-123.389 I},{x->-157.899+123.389 I},{x->-121.528-216.827 I},{x->-121.528+216.827 I},{x->-25.0454-326.078 I},{x->-25.0454+326.078 I},{x->201.816 -438.138 I},{x->201.816 +438.138 I}}},{n=15,{{x->-88.8262},{x->-39.4784},{x->-9.8696},{x->-182.992-78.1152 I},{x->-182.992+78.1152 I},{x->-171.703-163.08 I},{x->-171.703+163.08 I},{x->-165.48-7.46363 I},{x->-165.48+7.46363 I},{x->-122.56-265.614 I},{x->-122.56+265.614 I},{x->-6.42247-382.376 I},{x->-6.42247+382.376 I},{x->253.244 -497.508 I},{x->253.244 +497.508 I}}},{n=16,{{x->-157.311},{x->-88.8265},{x->-39.4784},{x->-9.8696},{x->-204.036-113.004 I},{x->-204.036+113.004 I},{x->-199.201-36.0916 I},{x->-199.201+36.0916 I},{x->-183.522-207.385 I},{x->-183.522+207.385 I},{x->-120.5-318.714 I},{x->-120.5+318.714 I},{x->16.5171 -442.471 I},{x->16.5171 +442.471 I},{x->310.485 -559.825 I},{x->310.485 +559.825 I}}},{n=17,{{x->-214.675},{x->-157.992},{x->-88.8264},{x->-39.4784},{x->-9.8696},{x->-224.949-66.4494 I},{x->-224.949+66.4494 I},{x->-223.871-152.559 I},{x->-223.871+152.559 I},{x->-193.117-256.229 I},{x->-193.117+256.229 I},{x->-115.162-376.033 I},{x->-115.162+376.033 I},{x->43.9074 -506.264 I},{x->43.9074 +506.264 I},{x->373.612 -625.004 I},{x->373.612 +625.004 I}}},{n=18,{{x->-157.905},{x->-88.8264},{x->-39.4784},{x->-9.8696},{x->-251.009-101.088 I},{x->-251.009+101.088 I},{x->-242.211-196.776 I},{x->-242.211+196.776 I},{x->-236.652-24.6632 I},{x->-236.652+24.6632 I},{x->-200.273-309.537 I},{x->-200.273+309.537 I},{x->-106.376-437.482 I},{x->-106.376+437.482 I},{x->75.8692 -573.665 I},{x->75.8692 +573.665 I},{x->442.693 -692.964 I},{x->442.693 +692.964 I}}},{n=19,{{x->-241.088},{x->-157.914},{x->-88.8264},{x->-39.4784},{x->-9.8696},{x->-276.183-140.608 I},{x->-276.183+140.608 I},{x->-268.924-52.178 I},{x->-268.924+52.178 I},{x->-258.82-245.593 I},{x->-258.82+245.593 I},{x->-204.797-367.237 I},{x->-204.797+367.237 I},{x->-93.9902-502.977 I},{x->-93.9902+502.977 I},{x->112.512 -644.588 I},{x->112.512 +644.588 I},{x->517.79 -763.635 I},{x->517.79 +763.635 I}}},{n=20,{{x->-285.946},{x->-248.158},{x->-157.914},{x->-88.8264},{x->-39.4784},{x->-9.8696},{x->-300.21-184.776 I},{x->-300.21+184.776 I},{x->-299.725-87.5671 I},{x->-299.725+87.5671 I},{x->-273.486-298.955 I},{x->-273.486+298.955 I},{x->-206.512-429.255 I},{x->-206.512+429.255 I},{x->-77.8649-572.438 I},{x->-77.8649+572.438 I},{x->153.936 -718.954 I},{x->153.936 +718.954 I},{x->598.958 -836.949 I},{x->598.958 +836.949 I}}}}

dingjifen

kastin 发表于 2020-1-19 23:03
详见杨翠红、朱思铭、梁肇军的论文《多项式代数方程根的完全分类及其应用》。

判定一个具体方程有无虚根，此论文没毛病。问题在于——当n为所有正整数时，方程有无虚根，此论文没法证明。

dingjifen

zeroieme 发表于 2020-1-20 17:37
{{n=1,{{x->-2.}}},{n=2,{{x->-9.4641},{x->-2.5359}}},{n=3,{{x->-2.4646},{x->-13.7677+10.1285 I},{ ...

看不明白是啥意思？  能否具体说明一下？