Algebraic tsis sib npaug lossis lawv cov tshuab nrog cov txiaj ntsig sib txawv uas nws cov kev daws teeb meem tau nrhiav hauv cov lej lossis cov lej. Raws li txoj cai, tus naj npawb ntawm cov tsis paub hauv Diophantine sib npaug yog ntau dua. Yog li, lawv tseem hu ua indefinite inequalities. Hauv kev ua lej niaj hnub no, lub tswv yim saum toj no yog siv rau kev sib npaug ntawm algebraic uas nws cov kev daws teeb meem yog nrhiav hauv algebraic integers ntawm qee qhov txuas ntxiv ntawm Q-rational variables, teb ntawm p-adic variables, thiab lwm yam.
Lub hauv paus ntawm cov kev tsis sib xws no
Txoj kev kawm ntawm Diophantine sib npaug yog nyob ntawm ciam teb ntawm tus lej txoj kev xav thiab algebraic geometry. Nrhiav cov kev daws teeb meem hauv cov lej sib txawv yog ib qho ntawm cov teeb meem lej qub tshaj plaws. Twb yog thaum pib ntawm lub xyoo txhiab xyoo BC. cov Babylonians thaum ub tau tswj hwm los daws cov kab ke ntawm kev sib npaug nrog ob yam tsis paub. Cov ceg ntawm kev ua lej no tau vam meej tshaj plaws hauv tebchaws Greece thaum ub. Tus lej ntawm Diophantus (ca. 3rd xyoo pua AD) yog qhov tseem ceeb thiab qhov tseem ceeb uas muaj ntau hom thiab cov kab ke sib npaug.
Nyob hauv phau ntawv no, Diophantus tau pom ntau txoj hauv kev los kawm txog qhov tsis sib xws ntawm qhov thib ob thiab thib pebcov degrees uas tau tsim muaj nyob rau hauv lub xyoo pua puv 19. Lub creation ntawm txoj kev xav ntawm tus lej rational los ntawm tus kws tshawb fawb ntawm tim Nkij teb chaws thaum ub tau coj mus rau kev tsom xam cov kev daws teeb meem rau cov txheej txheem tsis kawg, uas tau ua raws li hauv nws phau ntawv. Txawm hais tias nws txoj haujlwm muaj cov kev daws teeb meem rau qhov sib npaug ntawm Diophantine, nws muaj laj thawj ntseeg tias nws kuj tau paub ntau txoj hauv kev.
Kev kawm txog qhov tsis sib xws no feem ntau cuam tshuam nrog cov teeb meem loj. Vim lub fact tias lawv muaj polynomials nrog integer coefficients F (x, y1, …, y). Raws li qhov no, cov lus xaus tau kos tias tsis muaj ib qho algorithm uas tuaj yeem siv los txiav txim siab rau ib qho twg x seb qhov sib npaug F (x, y1, …., y ). Qhov xwm txheej no daws tau y1, …, y . Piv txwv ntawm cov polynomials tuaj yeem sau tau.
Qhov tsis sib xws yooj yim
ax + by=1, qhov twg a thiab b yog cov lej sib npaug thiab cov lej tseem ceeb, nws muaj ntau qhov kev tua (yog tias x0, y0 qhov tshwm sim yog tsim, ces tus khub ntawm qhov sib txawv x=x0 + b thiab y=y0 -an, qhov twg n yog arbitrary, tseem yuav suav tias yog qhov tsis sib xws). Lwm qhov piv txwv ntawm Diophantine equations yog x2 + y2 =z2. Cov kev daws teeb meem zoo ntawm qhov tsis sib xws yog qhov ntev ntawm cov sab me me x, y thiab txoj cai daim duab peb sab, nrog rau lub hypotenuse z nrog cov zauv sab qhov ntev. Cov lej no hu ua Pythagorean cov lej. Txhua triplets nrog kev hwm rau prime qhiasaum toj no variables yog muab los ntawm x=m2 – n2, y=2mn, z=m2+ n2, qhov twg m thiab n yog tus lej thiab tus lej thawj (m>n>0).
Diophantus hauv nws qhov kev tshawb nrhiav lej rau kev sib piv (tsis tas yuav tsum muaj) cov kev daws teeb meem tshwj xeeb ntawm nws qhov tsis sib xws. Ib txoj kev xav dav dav rau kev daws qhov sib npaug diophantine ntawm thawj qib yog tsim los ntawm C. G. Baschet nyob rau hauv lub xyoo pua 17th. Lwm cov kws tshawb fawb thaum pib ntawm lub xyoo pua 19th feem ntau kawm txog qhov tsis sib xws xws li ax2 +bxy + cy2 + dx +ey +f=0, qhov twg a, b, c, d, e, thiab f yog general, heterogeneous, nrog ob yam tsis paub ntawm qib thib ob. Lagrange siv cov feem ntau txuas ntxiv hauv nws txoj kev kawm. Gauss rau quadratic forms tsim ib qho kev xav dav dav hauv qee hom kev daws teeb meem.
Nyob rau hauv txoj kev tshawb fawb txog qhov tsis sib xws ntawm qib thib ob, kev nce qib tseem ceeb tsuas yog ua rau xyoo pua 20th. A. Thue pom tias Diophantine equation a0x + a1xn- 1 y +…+a y =c, where n≧3, a0, …, a , c yog integers, thiab a0tn + … + a tsis tuaj yeem muaj tus lej tsis kawg ntawm cov kev daws teeb meem. Txawm li cas los xij, Thue txoj kev tsis tau tsim kom raug. A. Baker tsim cov theorems zoo uas muab kev kwv yees ntawm kev ua tau zoo ntawm qee qhov sib npaug ntawm hom no. BN Delaunay tau thov lwm txoj hauv kev tshawb nrhiav siv tau rau cov chav nqaim ntawm cov kev tsis sib xws no. Tshwj xeeb, daim ntawv ax3 + y3 =1 yog daws tau los ntawm txoj kev no.
Diophantine sib npaug: txoj kev daws teeb meem
Txoj kev xav ntawm Diophantus muaj ntau cov lus qhia. Yog li, qhov teeb meem paub zoo hauv qhov system no yog qhov kev xav tias tsis muaj qhov tsis tseem ceeb ntawm qhov sib npaug ntawm Diophantine xn + y =z n if n ≧ 3 (Fermat's question). Txoj kev tshawb fawb ntawm cov lej ua tiav ntawm qhov tsis sib xws yog qhov dav dav ntawm qhov teeb meem ntawm Pythagorean triplets. Euler tau txais qhov kev daws teeb meem zoo ntawm Fermat qhov teeb meem rau n=4. Los ntawm qhov txiaj ntsig ntawm qhov txiaj ntsig no, nws hais txog cov pov thawj ntawm cov lej uas ploj lawm, tsis yog xoom cov kev tshawb fawb ntawm cov kab zauv yog n yog tus lej lej khib.
Kev kawm txog qhov kev txiav txim siab tsis tau ua tiav. Cov teeb meem nrog nws cov kev siv yog muaj feem xyuam rau qhov tseeb hais tias qhov yooj yim factorization nyob rau hauv lub nplhaib ntawm algebraic integers tsis yog tshwj xeeb. Txoj kev xav ntawm divisors nyob rau hauv no system rau ntau chav kawm ntawm prime exponents n ua rau nws muaj peev xwm mus xyuas kom meej lub validity ntawm Fermat lub theorem. Yog li, txoj kab sib npaug Diophantine nrog ob qhov tsis paub yog ua tiav los ntawm txoj kev thiab txoj hauv kev uas twb muaj lawm.
Nyob thiab hom haujlwm piav qhia
Arithmetic of rings of algebraic integers kuj tseem siv tau rau ntau yam teeb meem thiab kev daws teeb meem ntawm Diophantine sib npaug. Piv txwv li, cov qauv no tau siv thaum ua tiav qhov tsis sib xws ntawm daim ntawv N(a1 x1 +…+ a x)=m, qhov twg N(a) yog tus qauv ntawm a, thiab x1, …, xn pom muaj qhov sib txawv ntawm qhov sib txawv. Chav kawm no suav nrog Pell equation x2–dy2=1.
Tus nqi a1, …, a uas tshwm sim, cov kab zauv no muab faib ua ob hom. Thawj hom - hu ua tiav cov ntawv - suav nrog cov kev sib npaug uas ntawm ib qho muaj m linearly ywj siab tus lej hla ntawm qhov kev sib txawv ntawm qhov sib txawv Q, qhov twg m=[Q(a1, …, a):Q], nyob rau hauv uas muaj ib tug degree ntawm algebraic exponents Q (a1, …, a ) dhau Q. Cov hom tsis tiav yog cov nyob rau hauv uas qhov siab tshaj ntawm ai tsawg dua m.
Cov ntawv puv yog yooj yim dua, lawv txoj kev kawm tiav, thiab txhua qhov kev daws teeb meem tuaj yeem piav qhia. Hom thib ob, hom tsis tiav, yog qhov nyuaj dua, thiab kev txhim kho ntawm qhov kev xav no tseem tsis tau tiav. Cov kev sib npaug no tau kawm siv Diophantine approximations, uas suav nrog qhov tsis sib xws F (x, y)=C, qhov twg F (x, y) yog qhov tsis tuaj yeem, homogeneous polynomial ntawm degree n≧3. Yog li, peb tuaj yeem xav tias yi →∞. Raws li, yog tias yi yog qhov loj txaus, ces qhov tsis sib xws yuav tawm tsam lub theorem ntawm Thue, Siegel thiab Roth, los ntawm qhov nws ua raws li F(x, y)=C, qhov twg F yog. ib daim ntawv ntawm qib peb los yog siab dua, qhov irreducible tsis tuaj yeem muaj cov kev daws teeb meem tsis kawg.
Yuav daws qhov sib npaug Diophantine li cas?
piv txwv no yog ib chav nqaim heev ntawm txhua tus. Piv txwv li, txawm lawv qhov yooj yim, x3 + y3 + z3=N, thiab x2 +y 2 +z2 +u2 =N tsis suav nrog hauv chav kawm no. Txoj kev tshawb fawb txog kev daws teeb meem yog ib qho ua tib zoo kawm ceg ntawm Diophantine equations, qhov twg lub hauv paus yog qhov sawv cev los ntawm cov ntaub ntawv quadratic ntawm cov lej. Lagrangetsim ib txoj kev xav uas hais tias kev ua tiav muaj nyob rau txhua qhov ntuj N. Txhua tus lej tuaj yeem sawv cev raws li qhov sib npaug ntawm peb lub xwmfab (Gauss's theorem), tab sis nws yuav tsum tsis yog ntawm daim ntawv 4a (8K- 1), qhov twg a thiab k tsis yog cov lej lej tsis zoo.
Rational or integral solutions to a system of Diophantine equation of type F (x1, …, x)=a, qhov twg F (x 1, …, x) yog ib daim ntawv plaub npaug nrog cov lej coefficients. Yog li, raws li Minkowski-Hasse theorem, qhov tsis sib xws ∑aijxixj=b ij thiab b yog qhov laj thawj, muaj kev daws teeb meem hauv cov lej tiag thiab p-adic rau txhua tus lej p tsuas yog tias nws daws tau hauv cov qauv no.
Vim qhov teeb meem tshwm sim, kev kawm ntawm cov lej nrog cov ntawv tsis txaus ntseeg ntawm qib peb qib thiab siab dua tau kawm kom tsawg dua. Txoj kev ua haujlwm tseem ceeb yog txoj hauv kev ntawm trigonometric sums. Hauv qhov no, tus naj npawb ntawm cov kev daws teeb meem rau qhov sib npaug yog sau meej meej hauv cov ntsiab lus ntawm Fourier integral. Tom qab ntawd, cov txheej txheem ib puag ncig yog siv los nthuav qhia cov naj npawb ntawm kev ua tiav ntawm qhov tsis sib xws ntawm cov kev sib raug zoo sib xws. Txoj kev ntawm trigonometric sums nyob ntawm qhov algebraic nta ntawm qhov tsis sib xws. Muaj ntau ntau txoj hauv kev los daws cov kab sib npaug Diophantine.
Diophantine tsom xam
Department of lej, qhov kev kawm yog kev kawm ntawm cov ntsiab lus thiab cov kev daws teeb meem ntawm cov kab ke ntawm kev sib npaug ntawm algebra los ntawm cov txheej txheem ntawm geometry, los ntawm tib yamkheej kheej. Nyob rau hauv lub thib ob ib nrab ntawm lub xyoo pua puv 19, qhov tshwm sim ntawm tus xov tooj txoj kev xav tau coj mus rau txoj kev tshawb no ntawm Diophantine equations los ntawm ib tug arbitrary teb nrog coefficients, thiab cov kev daws teeb meem yog xam nyob rau hauv nws los yog nyob rau hauv nws rings. Lub kaw lus ntawm algebraic functions tsim nyob rau hauv parallel nrog cov zauv. Qhov yooj yim piv ntawm ob, uas tau hais los ntawm D. Hilbert thiab, tshwj xeeb tshaj yog, L. Kronecker, tau coj mus rau kev tsim kho ntawm ntau lub tswv yim lej, uas feem ntau hu ua ntiaj teb.
Qhov no yog qhov tshwj xeeb tshaj yog pom tau yog tias cov haujlwm algebraic nyob rau hauv kev kawm dhau ib thaj tsam ntawm qhov tsis tu ncua yog ib qho sib txawv. Cov tswv yim xws li chav kawm kev tshawb xav, kev faib tawm, thiab kev sib cais thiab cov txiaj ntsig yog ib qho piv txwv zoo ntawm cov saum toj no. Qhov kev xav no tau txais kev pom zoo nyob rau hauv cov txheej txheem ntawm Diophantine tsis sib xws tsuas yog tom qab ntawd, thiab kev tshawb fawb tsis yog tsuas yog nrog cov lej sib npaug, tab sis kuj nrog cov coefficients uas ua haujlwm, pib tsuas yog xyoo 1950. Ib qho kev txiav txim siab hauv txoj hauv kev no yog kev txhim kho algebraic geometry. Kev kawm tib lub sijhawm ntawm cov lej thiab cov haujlwm, uas tshwm sim raws li ob qhov tseem ceeb ntawm tib yam kev kawm, tsis yog tsuas yog muab cov txiaj ntsig zoo nkauj thiab kev ntseeg siab, tab sis coj mus rau kev sib koom ua ke ntawm ob lub ntsiab lus.
Nyob rau hauv algebraic geometry, qhov kev xav ntawm ntau yam yog hloov los ntawm cov txheej txheem tsis sib xws ntawm qhov tsis sib xws ntawm qhov muab K, thiab lawv cov kev daws teeb meem yog hloov los ntawm cov ntsiab lus muaj txiaj ntsig nrog cov txiaj ntsig hauv K lossis hauv nws qhov kev ncua ntev. Ib qho tuaj yeem hais tau tias qhov teeb meem tseem ceeb ntawm Diophantine geometry yog kev kawm txog cov ntsiab lus rationalntawm ib qho algebraic teeb X(K), thaum X yog qee tus lej hauv daim teb K. Integer execution muaj lub ntsiab lus geometric hauv linear Diophantine equations.
Kev tsis sib xws kev tshawb fawb thiab kev xaiv ua tiav
Thaum kawm txog kev xav (lossis ib qho) cov ntsiab lus ntawm algebraic ntau yam, thawj qhov teeb meem tshwm sim, uas yog lawv lub neej. Hilbert qhov teeb meem thib kaum yog tsim los ua qhov teeb meem ntawm kev nrhiav ib txoj hauv kev los daws qhov teeb meem no. Nyob rau hauv tus txheej txheem ntawm kev tsim ib tug meej lub ntsiab lus ntawm lub algorithm thiab tom qab nws tau ua pov thawj tias tsis muaj xws li executions rau ib tug loj tus naj npawb ntawm cov teeb meem, qhov teeb meem tau txais ib tug pom tseeb tsis zoo tshwm sim, thiab cov lus nug uas nthuav tshaj plaws yog lub ntsiab lus ntawm cov chav kawm ntawm Diophantine sib npaug. rau qhov system saum toj no muaj nyob. Txoj hauv kev zoo tshaj plaws, los ntawm qhov pom ntawm algebraic, yog lub hauv paus ntsiab lus hu ua Hasse: thawj daim teb K tau kawm ua ke nrog nws cov kev ua tiav Kv tshaj txhua qhov kev kwv yees. Txij li thaum X(K)=X(Kv) yog qhov tsim nyog rau lub neej, thiab K taw tes rau hauv tus account tias cov teeb X(Kv) tsis yog khoob rau txhua tus v.
Qhov tseem ceeb nyob ntawm qhov tseeb tias nws ua ke ob qho teeb meem. Qhov thib ob yog ntau yooj yim, nws yog solvable los ntawm ib tug paub algorithm. Hauv cov xwm txheej tshwj xeeb uas qhov ntau yam X yog qhov projective, Hansel's lemma thiab nws cov generalizations ua kom txo tau ntxiv: qhov teeb meem tuaj yeem txo qis rau kev kawm txog cov ntsiab lus muaj txiaj ntsig dhau ntawm thaj chaw finite. Tom qab ntawd nws txiav txim siab los tsim ib lub tswv yim los ntawm kev tshawb fawb zoo ib yam lossis txoj hauv kev zoo dua.
Xeemib qho kev txiav txim siab tseem ceeb yog tias cov teeb tsa X (K v) tsis yog qhov khoob rau txhua tus tab sis tus naj npawb ntawm v, yog li tus naj npawb ntawm cov xwm txheej ib txwm raug txiav thiab lawv tuaj yeem kuaj tau zoo. Txawm li cas los xij, Hasse lub hauv paus ntsiab lus tsis siv rau kev kawm nkhaus. Piv txwv li, 3x3 + 4y3=5 muaj cov ntsiab lus hauv txhua qhov p-adic tus lej thiab nyob rau hauv qhov system ntawm tus lej tiag, tab sis tsis muaj cov ntsiab lus rational.
Txoj kev no tau ua qhov pib rau kev tsim lub tswv yim piav qhia txog cov chav kawm ntawm qhov chaw tseem ceeb ntawm Abelian ntau yam los ua "kev sib txawv" los ntawm Hasse txoj cai. Nws tau piav qhia txog cov qauv tshwj xeeb uas tuaj yeem cuam tshuam nrog txhua qhov manifold (Tate-Shafarevich pawg). Qhov teeb meem tseem ceeb ntawm txoj kev xav nyob hauv qhov tseeb tias cov txheej txheem rau kev suav cov pab pawg yog qhov nyuaj kom tau txais. Lub tswv yim no kuj tau txuas ntxiv mus rau lwm chav kawm ntawm algebraic ntau yam.
Tshawb nrhiav cov algorithm kom ua tiav qhov tsis sib xws
Lwm lub tswv yim heuristic siv hauv kev kawm ntawm Diophantine equations yog tias yog tias tus lej ntawm qhov hloov pauv tau koom nrog hauv cov txheej txheem tsis sib xws loj, ces lub kaw lus feem ntau muaj kev daws teeb meem. Txawm li cas los xij, qhov no nyuaj heev los ua pov thawj rau txhua qhov xwm txheej tshwj xeeb. Txoj hauv kev dav dav rau cov teeb meem ntawm hom no siv cov lej ntsuas kev xav thiab yog raws li kev kwv yees rau cov lej trigonometric. Txoj kev no yog thawj zaug siv rau cov kev sib npaug tshwj xeeb.
Txawm li cas los xij, tom qab ntawd nws tau ua pov thawj nrog nws txoj kev pab tias yog tias daim ntawv ntawm qhov khib nyiab yog F, hauv dthiab n variables thiab nrog rational coefficients, ces n yog loj txaus piv rau d, yog li qhov projective hypersurface F=0 muaj ib tug rational point.. Qhov no tsuas yog ua pov thawj rau cov ntawv plaub. Cov teeb meem zoo sib xws tuaj yeem nug rau lwm qhov chaw thiab. Qhov teeb meem hauv nruab nrab ntawm Diophantine geometry yog cov qauv ntawm cov txheej txheem ntawm cov lej lossis cov ntsiab lus rational thiab lawv txoj kev kawm, thiab thawj lo lus nug yuav tsum tau qhia meej yog seb qhov teeb tsa no puas yog. Hauv qhov teeb meem no, qhov xwm txheej feem ntau muaj qhov txwv tsis pub ua yog tias qhov ntsuas ntawm qhov system loj dua li tus lej ntawm qhov hloov pauv. Nov yog qhov kev xav hauv paus.
Kev tsis sib xws ntawm kab thiab nkhaus
Pab pawg X(K) tuaj yeem sawv cev raws li qhov ncaj qha ntawm cov qauv dawb ntawm qib r thiab pawg tsis kawg ntawm kev txiav txim n. Txij li thaum xyoo 1930s, cov lus nug ntawm seb cov lej no puas raug khi rau ntawm cov txheej txheem ntawm txhua qhov elliptic nkhaus hla ib thaj chaw K tau kawm. Qhov kev sib tw ntawm torsion n tau pom nyob rau hauv lub seventies. Muaj cov nkhaus ntawm arbitrary qib siab nyob rau hauv rooj plaub ua haujlwm. Nyob rau hauv tus lej, tseem tsis muaj lus teb rau lo lus nug no.
Thaum kawg, Mordell qhov kev xav tau hais tias tus lej ntawm cov ntsiab lus tseem ceeb yog qhov kawg rau qhov nkhaus ntawm genus g>1. Nyob rau hauv cov ntaub ntawv ua haujlwm, lub tswv yim no tau pom los ntawm Yu. I. Manin hauv xyoo 1963. Cov cuab yeej tseem ceeb siv los ua pov thawj finiteness theorems hauv Diophantine geometry yog qhov siab. Ntawm cov algebraic ntau yam, qhov ntev saum ib qho yog abelianmanifolds, uas yog ntau qhov sib txawv ntawm elliptic curves, tau kawm kom meej tshaj plaws.
A. Weil generalized lub theorem ntawm lub finiteness ntawm tus naj npawb ntawm generators ntawm ib pab pawg neeg ntawm rational ntsiab lus rau Abelian ntau yam ntawm txhua qhov ntev (lub tswv yim Mordell-Weil), ncua nws. Hauv xyoo 1960, qhov kev xav ntawm Birch thiab Swinnerton-Dyer tau tshwm sim, txhim kho qhov no thiab pab pawg thiab zeta ua haujlwm ntawm manifold. Cov ntaub ntawv pov thawj txhawb qhov kev xav no.
Kev daws teeb meem
Qhov teeb meem ntawm kev nrhiav cov algorithm uas tuaj yeem siv los txiav txim seb puas muaj qhov sib npaug Diophantine muaj kev daws teeb meem. Ib qho tseem ceeb ntawm qhov teeb meem tau tshwm sim yog kev tshawb nrhiav rau txoj hauv kev thoob ntiaj teb uas yuav tsim nyog rau kev tsis sib xws. Xws li ib txoj kev kuj yuav tso cai rau kev daws cov kab ke saum toj no, vim tias nws sib npaug rau P21 + ⋯ + P2k=0.p1=0, …, PK=0p=0, …, pK=0 lossis p21 + ⋯ + P2K=0. n12+⋯+pK2=0. Qhov teeb meem ntawm kev nrhiav ib txoj hauv kev zoo li no los nrhiav kev daws teeb meem rau cov kab tsis sib xws hauv cov lej tau tshwm sim los ntawm D. Gilbert.
Nyob rau thaum ntxov xyoo 1950, thawj cov kev tshawb fawb tau tshwm sim los ua pov thawj qhov tsis muaj nyob ntawm cov algorithm rau kev daws qhov sib npaug Diophantine. Lub sijhawm no, Davis qhov kev xav tau tshwm sim, uas tau hais tias txhua qhov kev suav sau kuj tseem yog tus kws tshawb fawb Greek. Vim hais tias piv txwv ntawm algorithmically undecidable sets paub, tab sis yog recursively enumerable. Nws ua raws li qhov kev xav ntawm Davis yog qhov tseeb thiab qhov teeb meem ntawm solvability ntawm cov kab zauv nomuaj kev ua tsis zoo.
Tom qab ntawd, rau Davis qhov kev xav, nws tseem yuav ua pov thawj tias muaj ib txoj hauv kev los hloov qhov tsis sib xws uas tseem (lossis tsis tau) tib lub sijhawm muaj kev daws teeb meem. Nws tau pom tias qhov kev hloov pauv ntawm Diophantine sib npaug yog ua tau yog tias nws muaj ob lub zog saum toj no: 1) nyob rau hauv ib qho kev daws ntawm hom no v ≦ uu; 2) rau ib qho k, muaj kev ua tiav nrog kev loj hlob ntxiv.
Ib qho piv txwv ntawm kab zauv Diophantine kab zauv ntawm chav kawm no ua tiav cov pov thawj. Qhov teeb meem ntawm lub neej ntawm ib qho algorithm rau kev daws teeb meem thiab kev lees paub ntawm cov kev tsis sib xws hauv cov lej rational tseem suav tias yog ib qho tseem ceeb thiab qhib lo lus nug uas tsis tau kawm txaus.
