(original) (raw)

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b(is)e(often)i (required)d(to)j(\014nd)e(the)i(probabilit)m(y)d(of)i(the)h(union)d(of) j(giv)m(en)f Fs(n)g Ft(ev)m(en)m(ts)244 1536 y Fs(A)312 1550 y Fr(1)351 1536 y Fs(;)15 b(:)g(:)g(:)i(;)e(A)621 1550 y Fq(n)669 1536 y Ft(.)41 b(The)30 b(answ)m(er)h(is)e(pro)m (vided,)h(of)h(course,)g(b)m(y)f(the)h(inclusion-exclusion)c(form)m (ula:)244 1649 y(Pr)o(\()p Fp([)p Fs(A)505 1663 y Fq(i)534 1649 y Ft(\))36 b(=)712 1585 y Fo(P)800 1672 y Fq(i)843 1649 y Ft(Pr\()p Fs(A)1044 1663 y Fq(i)1072 1649 y Ft(\))25 b Fp(\000)1227 1585 y Fo(P)1315 1672 y Fq(i<j)1446 0="" 1="" 2="" 3="" 4="" 5="" 34="" 36="" 39="" 43="" 48="" 55="" 59="" 65="" 67="" 72="" 74="" 80="" 82="" 86="" 100="" 105="" 108="" 110="" 111="" 124="" 150="" 178="" 181="" 197="" 201="" 208="" 220="" 222="" 231="" 244="" 254="" 266="" 273="" 276="" 282="" 296="" 303="" 325="" 329="" 340="" 342="" 366="" 369="" 381="" 384="" 394="" 395="" 404="" 407="" 416="" 431="" 457="" 461="" 475="" 486="" 498="" 511="" 512="" 513="" 540="" 561="" 567="" 569="" 578="" 597="" 603="" 611="" 614="" 661="" 667="" 686="" 697="" 701="" 706="" 711="" 712="" 721="" 760="" 763="" 766="" 774="" 779="" 785="" 786="" 791="" 796="" 798="" 803="" 818="" 837="" 838="" 852="" 923="" 926="" 955="" 968="" 971="" 990="" 1009="" 1021="" 1024="" 1028="" 1051="" 1086="" 1088="" 1109="" 1115="" 1125="" 1133="" 1148="" 1156="" 1171="" 1183="" 1186="" 1196="" 1207="" 1224="" 1231="" 1259="" 1267="" 1286="" 1304="" 1331="" 1338="" 1371="" 1375="" 1382="" 1385="" 1387="" 1390="" 1398="" 1402="" 1422="" 1424="" 1456="" 1500="" 1508="" 1523="" 1536="" 1542="" 1544="" 1548="" 1557="" 1563="" 1602="" 1617="" 1626="" 1642="" 1649="" 1663="" 1665="" 1668="" 1675="" 1680="" 1686="" 1687="" 1690="" 1699="" 1701="" 1707="" 1722="" 1724="" 1762="" 1766="" 1790="" 1796="" 1811="" 1831="" 1845="" 1858="" 1860="" 1875="" 1880="" 1883="" 1915="" 1916="" 1946="" 1950="" 1951="" 1954="" 1965="" 1988="" 1989="" 2034="" 2036="" 2090="" 2101="" 2119="" 2144="" 2155="" 2157="" 2193="" 2197="" 2211="" 2214="" 2215="" 2216="" 2227="" 2241="" 2253="" 2257="" 2261="" 2266="" 2277="" 2278="" 2287="" 2299="" 2311="" 2314="" 2317="" 2325="" 2327="" 2329="" 2340="" 2352="" 2361="" 2376="" 2377="" 2382="" 2397="" 2406="" 2408="" 2411="" 2412="" 2425="" 2439="" 2447="" 2449="" 2451="" 2460="" 2464="" 2473="" 2483="" 2509="" 2511="" 2518="" 2521="" 2533="" 2539="" 2552="" 2569="" 2584="" 2593="" 2594="" 2601="" 2608="" 2617="" 2625="" 2634="" 2640="" 2649="" 2661="" 2665="" 2675="" 2677="" 2679="" 2687="" 2713="" 2721="" 2730="" 2755="" 2758="" 2761="" 2778="" 2781="" 2786="" 2796="" 2797="" 2807="" 2824="" 2827="" 2833="" 2837="" 2838="" 2849="" 2853="" 2857="" 2858="" 2870="" 2885="" 2891="" 2902="" 2905="" 2907="" 2912="" 2914="" 2917="" 2932="" 2936="" 2937="" 2947="" 2951="" 2957="" 2959="" 2973="" 2980="" 2990="" 2996="" 3000="" 3007="" 3020="" 3021="" 3022="" 3023="" 3031="" 3032="" 3033="" 3034="" 3037="" 3048="" 3066="" 3074="" 3079="" 3089="" 3097="" 3113="" 3131="" 3136="" 3153="" 3158="" 3210="" 3217="" 3227="" 3232="" 3242="" 3248="" 3320="" 3328="" 3337="" 3352="" 3361="" 3375="" 3384="" 3394="" 3396="" 3409="" 3415="" 3423="" 3437="" 3441="" 3448="" 3458="" 3474="" 3488="" 3514="" 3529="" 3535="" 3544="" 3554="" 3561="" 3578="" 3587="" 3593="" 3601="" 3636="" 3639="" 3677="" 3682="" 3685="" 3692="" 3700="" 3702="" 3710="" 3713="" 3729="" 3734="" 3735="" 3744="" 3759="" 3766="" 3770="" 3785="" 3800="" 3801="" 3802="" 3813="" 3827="" 3830="" 3835="" 3842="" 3857="" 3861="" 3866="" 3884="" 3922="" 3926="" 3940="" 3941="" 3950="" 4039="" 4043="" 4071="" 4113="" 4116="" 4149="" 4152="" 4163="" 4174="" 4212="" 4237="" 4252="" 4257="" 4265="" 4269="" 4272="" 4283="" 4294="" 4356="" 4357="" 4359="" 4372="" 4377="" 4378="" 4392="" 4404="" 4415="" 4438="" 4452="" 4454="" 4475="" 4478="" 4490="" 4493="" 4499="" 4512="" 4514="" 4571="" 4598="" 4609="" 4656="" 4669="" 4670="" 4679="" 4692="" 4695="" 4700="" 4713="" 4718="" 4736="" 4751="" 4753="" 4800="" 4804="" 4812="" 4839="" 4847="" 4870="" 4900="" 4980="" 4995="" 5000="" 5070="" 5100="" 5349="" y="" ft(pr)o(\()p="" fs(a)1646="" fq(i)1699="" fp(\\)g="" fs(a)1853="" fq(j)1889="" ft(\))g="" fp(\006)f="" fs(:)15="" b(:)g(:)r="" ft(.)60="" b(unfortunately)-8="" b(,)38="" b(this)e(form)m(ula)g(has)="" y(exp)s(onen)m(tially)c(man)m(y)j(terms,)g(and)e(only)h="" (rarely)f(do)s(es)h(one)g(manage)i(to)e(carry)h(out)f(the)g(ex-)244="" y(act)41="" b(calculation.)69="" b(f)-8="" b(rom)40="" b(a)g(computational)g="" (p)s(oin)m(t)f(of)h(view,)i(\014nding)37="" b(the)j(probabilit)m(y)e(of)="" y(the)33="" b(union)e(is)h(an)g(in)m(tractable,)i(#p-hard)d="" (problem,)i(ev)m(en)g(in)e(v)m(ery)j(restricted)e(cases.)49="" b(this)244="" y(state)40="" b(of)f(a\013airs)g(mak)m(es)h(it)e="" (reasonable)h(to)h(seek)g(appro)m(ximate)f(solutions)e(that)j(are)f="" (com-)244="" y(putationally)h(feasible.)72="" b(a)m(ttempts)43="" b(to)f(\014nd)e(suc)m(h)h(appro)m(ximate)h(solutions)e(ha)m(v)m(e)j(a)e="" (long)244="" y(history)33="" b(starting)h(already)g(with)f(bo)s(ole)h="" ([1)q(].)53="" b(a)34="" b(recen)m(t)h(step)f(in)f(this)g(direction)g(w)m(as)="" i(tak)m(en)244="" y(b)m(y)27="" b(linial)d(and)i(nisan)g([4)q(])h(who)f="" (sough)m(t)h(appro)m(ximations)f(to)h(the)g(probabilit)m(y)e(of)i(the)g="" (union,)244="" y(giv)m(en)f(the)h(probabilities)c(of)k(all)f="" fs(j)5="" b="" ft(-wise)26="" b(in)m(tersections)g(of)h(the)g(ev)m(en)m(ts)h="" (for)e="" fs(j)31="" ft(=")25" b(1)p="" fs(;)15="" b(:)g(:)g(:)i(k)s="" ft(.)40="" b(they)244="" y(dev)m(elop)s(ed)33="" b(a)i(metho)s(d)e(to)i="" (appro)m(ximate)f(pr\()p="" fp([)p="" fs(a)1989="" fq(i)2017="" ft(\),)i(from)d(the)h(ab)s(o)m(v)m(e)i(data)e(with)f(an)h(addi-)="" y(tiv)m(e)29="" b(error)g(of)h(exp\()p="" fp(\000)p="" fs(o)s="" ft(\()p="" fs(k)s(=")1191" fp(p)p="" v="" x="" fs(n)p="" ft(\)\).)41="" b(in)28="" b(the)i(presen)m(t)f(article)g="" (w)m(e)g(dev)m(elop)g(an)h(expression)d(that)244="" y(can)33="" b(b)s(e)g(computed)g(in)f(p)s(olynomial)f(time,)j(that,)h(giv)="" m(en)e(the)h(sums)2637="" fo(p)2725="" fn(j)p="" fq(s)t="" fr(=")p" fq(j)2917="" ft(pr\()p="" fp(\\)3111="" fq(i)p="" fn(2)p="" fq(s)3232="" fs(a)3300="" fq(i)3329="" ft(\))f(for)244="" fs(j)f="" ft(,)32="" b(appro)m(ximates)f(pr\()p="" fs(a)1548="" fq(i)1576="" ft(\))g(with)f(an)h(additiv)m="" (e)g(error)f(of)i(exp)o(\()p="" fp(\000)2909="" ft(~)2899="" y(\012\()p="" fs(k)3050="" fr(2)3090="" fs(="n)p" ft(\)\).)44="" y(error)30="" b(is)f(optimal,)h(up)f(to)i(the)="" g(logarithmic)e(factor)i(implicit)d(in)h(the)2662="" y(~)2652="" y(\012)h(notation.)380="" y(the)21="" b(problem)e(of)i(en)="" m(umerating)f(satisfying)g(assignmen)m(ts)g(of)h(a)h(b)s(o)s(olean)e="" (form)m(ula)g(in)f(dnf)244="" y(form)35="" fs(f)47="" fp(_)735="" fq(m)735="" fr(1)801="" fs(c)866="" fq(i)930="" ft(is)h(an)h(instance)f(of)h(the)g="" (general)g(problem)e(that)i(had)f(b)s(een)g(extensiv)m(ely)244="" y(studied)f([7)q(].)57="" b(here)37="" fs(a)1035="" fq(i)1099="" ft(is)e(the)h(set)g(of)h(assignmen)m(ts)e(that)i="" (satisfy)e="" fs(c)2681="" fq(i)2709="" ft(,)j(and)d(pr)o(\()p="" fp(\\)3147="" fq(s)3269="" fs(a)3337="" fq(i)3366="" ft(\))f(=")244" fs(a)292="" fq(s)343="" ft(2)433="" fq(n)526="" ft(where)44="" fp(^)864="" fq(s)986="" fs(c)1051="" fq(i)1124="" ft(is)g(satis\014ed)g(b)m(y)h="" fs(a)1783="" fq(s)1879="" ft(assignmen)m(ts.)85="" b(judging)43="" b(from)i(the)g(general)244="" y(results,)40="" b(it)e(is)g(hard)g(to)h(exp)s(ect)g(a)h(decen)m(t)f(appro)m(ximation)f="" (of)h="" fs(f)13="" ft('s)39="" b(n)m(um)m(b)s(er)e(of)i(satisfying)244="" y(assignmen)m(ts,)f(without)d(kno)m(wledge)i(of)f(the)h(n)m(um)m="" (b)s(ers)e="" fs(a)2264="" fq(s)2351="" ft(for,)j(sa)m(y)-8="" b(,)39="" b(all)d(cardinalities)e(1)h="" fp(\024)244="" y(j)p="" fs(s)5="" fp(j)48="" b(\024)521="" y(p)p="" fs(m)p="" ft(.)80="" b(quite)43="" b(surprisingly)-8="" b(,)44="" b(already)f(the)h(n)m(um)m(b)s(ers)e="" fs(a)2509="" fq(s)2604="" ft(o)m(v)m(er)j="" fp(j)p="" b(\024)f="" ft(log)q(\()p="" fs(n)29="" ft(+)g(1\))244="" fm(uniquely)j(determine)h(the)g(numb)-5="" b(er)33="" b(of)g(satisfying)g="" (assignments)h(for)d="" ft(.)380="" y(w)-8="" b(e)32="" b(p)s(oin)m(t)d(out)i(a)g(connection)f(b)s(et)m(w)m(een)h(our)f(w)m="" (ork)h(and)f(the)g(edge-reconstruction)h(con-)244="" y(jecture.)42="" b(finally)29="" b(w)m(e)i(discuss)e(other)i(sp)s(ecial)e="" (instances)h(of)h(the)g(problem,)f(e.g.,)i(computing)244="" y(p)s(ermanen)m(ts)k(of)i(0)p="" ft(1)38="" b(matrices,)h(ev)-5="" b(aluating)37="" b(c)m(hromatic)h(p)s(olynomials)c(of)="" j(graphs)g(and)f(for)244="" y(families)28="" b(of)j(ev)m(en)m(ts)g="" (whose)g(v)m(c)f(dimension)e(is)h(b)s(ounded.)p="" a="" fl(\003)149="" fk(rutgers)48="" b(univ)n(ersit)n(y)-7="" b(.)96="" b(w)-7="" b(ork)48="" b(supp)r(orted)f(in)h="" (part)g(b)n(y)f(a)h(gran)n(t)e(of)i(the)g(binational)f(israel-us)g="" (science)0="" y(f)-7="" b(oundation.)115="" fl(y)149="" fk(hebrew)27="" b(.)36="" b(ork)26="" b(supp)r(orted)g(in)h(part)g(b)n(y)f(a)g(gran)n(t)g(of)g(the)i="" (binational)d(israel-us)h(science)h(f)-7="" b(ounda-)0="" y(tion)28="" b(and)f(b)n(y)g(the)h(israel)f(science)g(f)-7="" fl(z)149="" fk(hebrew)28="" b(univ)n(ersit)n(y)1851="" fj(1)p="" eop="" %%page:="" bop="" fi(1)161="" b(in)l(tro)t(duction)0="" fj(let)23="" fh(a)238="" fr(1)277="" fh(;)17="" b(:)g(:)g(:)f(;)h(a)="" fq(n)638="" fj(b)s(e)23="" b(ev)m(en)m(ts)i(in)c(a)i="" (probabilit)m(y)d(space.)41="" b(the)24="" b(inclusion-exclusion)c(form)m="" (ula)h(expresses)0="" b(probabilit)m(y)e(of)h(their)g(union:)="" y(pr\()p="" fg([)p="" fh(a)623="" fq(i)652="" fj(\))27="" b(=")821" fo(x)869="" fq(i)957="" fj(pr)q(\()p="" fh(a)1173="" fq(i)1201="" fj(\))22="" fg(\000)1360="" fo(x)1365="" fq(i<j)1497="" fj(pr\()p="" fh(a)1712="" fq(i)1763="" fg(\\)g="" fh(a)1924="" fq(j)1961="" fj(\))g(+)2161="" fo(x)2119="" fq(i<j)t(<k)2340="" fh(a)2556="" fq(i)2606="" fg(\\)h="" fh(a)2768="" fq(j)2826="" fh(a)2988="" fq(k)3031="" fj(\))f="" fg(\007)g="" fh(:)17="" b(:)g(:)28="" fj(=")1238" y(=")1419" fo(x)1342="" fr([)p="" fq(n)p="" fr(])p="" fn(\023)p="" fn(6)p="" fn(;)1616="" fj(\()p="" fg(\000)p="" fj(1\))1818="" fn(j\000)p="" fr(1)2015="" fg(\\)2223="" fq(s)2345="" fh(a)2418="" fq(i)2446="" fj(\))p="" fh(:)0="" fj(if)34="" b(an)m(y)h(of)f(the)h="" (2)617="" fq(n)687="" fg(\000)24="" fj(1)34="" b(terms)g(is)g="" (unkno)m(wn,)i(it)e(is)g(not)g(p)s(ossible,)g(in)f(general,)i(to)f="" (exactly)g(ev)-5="" b(aluate)0="" fh(a)281="" fq(i)310="" fj(\).)68="" b(man)m(y)41="" b(in)m(v)m(estigators)g="" (\(e.g.)68="" b([2])40="" b(and)h(the)g(references)i(therein\))d(had)h(ask)m="" (ed)h(ho)m(w)g(w)m(ell)0="" fj(\))29="" b(can)g(b)s(e)g="" ff(appr)-5="" b(oximate)g(d)27="" fj(giv)m(en)i(only)f(partial)f(information)f(on)i="" (the)h(n)m(um)m(b)s(ers)h(pr\()p="" fg(\\)3461="" fq(s)3584="" fh(a)3657="" fq(i)3685="" fj(\).)146="" y(linial)42="" b(and)k(nisan)e([4])h(sho)m(w)m(ed)i="" (that)e(if)f(pr\()p="" fg(\\)1956="" fq(s)2078="" fh(a)2151="" fq(i)2180="" fj(\))g(is)h(kno)m(wn)h="" (whenev)m(er)i="" fg(j)p="" fh(s)6="" fg(j)48="" b(\024)h="" fh(k)s="" fj(,)f(then)0="" fj(\))25="" b(ma)m(y)f(b)s(e)i(b)s(e)f(appro)m(ximated)f(as)h="" (follo)m(ws:)38="" b(if)25="" fh(k)31="" fh(o)s="" fj(\()2294="" fg(p)p="" fh(n)p="" fj(\),)f(then)g(it)e(is)g(p)="" s(ossible)g(to)h(estimate)0="" y(the)38="" b(probabilit)m(y)e(of)h(the)h="" (union)f(to)h(within)e(an)i(additiv)m(e)f(error)h(of)f="" fj(\(1)24="" fg(\000)j="" fh(k)2902="" fr(2)2941="" fh(="n)p" fj(\).)59="" b(moreo)m(v)m(er,)40="" b(for)0="" fh(k)i="" fj(\()325="" fj(\),)h(this)f(is)e(also)h(essen)m(tially)g="" (optimal.)61="" b(ho)m(w)m(ev)m(er,)44="" b(for)39="" b(larger)f="" fj(,)k([4)o(])e(fails)e(to)h(pro)m(vide)0="" y(a)h(full)f(answ)m(er.)68="" b(a)41="" b(metho)s(d)e(of)h(appro)m(ximation)f="" (whic)m(h)i(is)e(dev)m(elop)s(ed)j(in)d(that)i(pap)s(er)f(o\013ers)h="" (an)0="" y(appro)m(ximation)32="" b(to)j(within)e="" fh(e)1124="" fn(\000)p="" fr(\012\()p="" fq(k)r(=")1331" fn(p)p="" fr(\))1499="" fj(of)h(the)h(truth.)50="" b(this)34="" b(degree)i(of)e(appro)m(ximation)e(has)j(b)s(een)0="" y(sho)m(wn)26="" b(\(ibid.\))39="" b(to)24="" b(b)s(e)g(sub)s(optimal.)38="" b(this)25="" b(problem)e(is)g(resolv)m(ed)j(in)d(the)i(presen)m(t)h="" (article:)37="" b(giv)m(en)25="" b(the)0="" y(n)m(um)m(b)s(ers)387="" fo(p)475="" fq(j)669="" fg(\\)877="" fq(s)999="" fh(a)1072="" fq(i)1100="" fj(\))i(for)e(all)f="" fh(j)34="" fg(\024)28="" fj(,)g(w)m(e)f(can)g(appro)m(ximate)e="" (the)i(probabilit)m(y)d(of)i(the)h(union)0="" y(to)35="" b(within)f(an)g(additiv)m(e)h(error)g(of)f="" fh(e)1341="" fn(\000)1404="" fr(~)1396="" fq(k)1513="" fe(2)1548="" fq(="n)p" fr(\))1657="" fj(.)51="" b(moreo)m(v)m(er,)37="" b(the)f(appro)m(ximation)c(can)k(b)="" s(e)f(computed)0="" y(in)f(p)s(olynomial)d(time.)49="" b(the)36="" b(result)f(is)f(tigh)m(t)g(in)h(the)g(sense)i(that)d(giv)m(en)="" h(the)g(n)m(um)m(b)s(ers)h(pr)q(\()p="" fg(\\)3489="" fq(s)3611="" fh(a)3684="" fq(i)3712="" fj(\))0="" y(for)d(all)e="" fg(j)28="" b(\024)i="" fj(,)k(it)e(is)h(in)f(general)h(imp)s(ossible)e(to)i="" (appro)m(ximate)g(pr\()p="" fh(a)2748="" fq(i)2777="" fj(\))g(to)g(within)f(an)i(additiv)m(e)0="" y(error)c(smaller)e(than)j="" fh(e)841="" fn(\000)912="" fr(~)896="" fq(o)q="" fr(\()p="" fq(k)1017="" fe(2)1052="" fr(\))1192="" fj(\(regardless)f(of)g="" (the)h(computational)d(complexit)m(y)h(in)m(v)m(olv)m(ed)i(in)f(the)0="" y(appro)m(ximation\).)146="" y(the)39="" b(problem)e(of)g(en)m="" (umerating)g(the)h(satisfying)f(assignmen)m(ts)h(of)f(a)h(dnf)e(form)m="" (ula)g(is)i(an)f(in-)0="" y(stance)43="" b(of)e(the)i(general)e="" (problem.)70="" b(already)42="" b(this)f(sp)s(ecial)g(case)i(is)e(kno)m(wn)j="" (to)d(b)s(e)h(complete)f(for)0="" y(the)c(class)f(#p)i(.)e(muc)m(h)i="" (atten)m(tion)e(has)h(b)s(een)g(giv)m(en)f(to)h(e\016cien)m(t)g="" (algorithms)d(for)i="" b(oximating)0="" fj(this)33="" b(n)m(um)m(b)s(er,)h(see)g([7])f(and)h(the)f(references)j(therein.)45="" b(t)-8="" b(o)34="" b(put)f(the)h(dnf)e(problem)g(in)h(the)h(general)0="" y(con)m(text)i(of)e(our)g(problem,)g(let)f(the)i(probabilit)m(y)e="" (space)i(b)s(e)g="" fg(f)p="" fj(0)p="" fg(g)2498="" fq(n)2578="" fj(under)35="" b(uniform)e(distribution.)0="" y(asso)s(ciated)38="" b(with)f(ev)m(ery)j(clause)e(in)e(the)j(dnf)d="" (form)m(ula)g(is)h(the)h(ev)m(en)m(t)i(that)d(this)h(clause)g(is)f="" (satis-)0="" y(\014ed.)44="" b(eac)m(h)34="" b(suc)m(h)g(ev)m(en)m(t)h(is,)="" d(in)g(fact,)g(a)h(sub)s(cub)s(e)h(of)e(the)h="" fj(-dimensional)d(cub)s(e.)44="" b(or)32="" b(this)g(problem)0="" y(something)38="" b(quite)h(remark)-5="" b(able)37="" b(happ)s(ens:)57="" b(while)38="" b(an)m(y)i(decen)m(t)g(appro)m(ximation)d(for)h(the)i="" (general)0="" y(problem)35="" b(requires)j(information)33="" b(on)j(\012\()1533="" fh(m)p="" fj(\)-wise)g(in)m(tersections,)j(here)e(the)g(answ)m(er)h="" (is)e="" ff(uniquely)0="" y(determine)-5="" b(d)32="" fj(from)f(the)i="" (probabilities)d(of)i="" fg(\024)c="" fj(\(log)16="" fh(n)23="" fj(+)f(1\)-wise)32="" b(in)m(tersections.)146="" y(our)i(metho)s(ds)f="" (o\013er)h(also)e(some)i(new)g(insigh)m(t)f(in)m(to)f(the)i="" (edge{reconstruction)h(problem)d(from)0="" y(graph)42="" b(theory)-8="" b(.)74="" b(in)42="" b(particular)f(m)s(\177)-51="" b(uller's)41="" b([8])h(theorem)h(can)f(b)s(e)h(repro)m(v)m(ed)h(and)e="" (put)h(in)f(a)g(more)0="" y(general)32="" b(con)m(text)i(that)e(ma)m(y)h="" (p)s(ossibly)f(lead)g(to)g(further)h(progress)g(on)g(this)f="" (conjecture.)146="" b(e)27="" b(then)h(p)s(oin)m(t)e(out)g(that)h="" (calculating)d(0)p="" fj(1)26="" b(p)s(ermanen)m(ts)i(ma)m(y)e="" (also)g(b)s(e)h(view)m(ed)h(as)f(an)g(instance)0="" y(of)f(the)g(general)g(problem,)h(and)f(similarly)c(the)27="" b(problem)e(of)h(computing)f(c)m(hromatic)g(p)s(olynomials)e(of)1851="" y(2)p="" fj(graphs.)45="" b(some)32="" b(commen)m(ts)h(are)g(made)g="" (on)g(the)g(p)s(ossibilit)m(y)e(of)h(getting)g(estimates)h(for)f(these)="" j(cases)0="" y(that)j(are)f(b)s(etter)i(than)f(those)g(ac)m(hiev)-5="" b(in)g(the)i(general)e(case.)60="" b(finally)35="" b(w)m(e)k(deriv)m(e)g(a)e(theorem)0="" y(similar)27="" b(to)k(the)g(one)g(for)f(dnf)g(form)m(ulas,)g(in)g(case)i(the)f(v)m(c)h="" (dimension)d(of)h(our)h(family)d(of)i(ev)m(en)m(ts)j(is)0="" y(b)s(ounded.)0="" fi(2)161="" b(near-tigh)l(t)78="" b(appro)l(ximations)f(for)i(general)f(inclusion-)242="" y(exclusion)0="" fj(the)34="" b(main)c(result)j(of)f(this)g="" (section)h(is:)0="" fd(theorem)k(2.1)-8="" b(:)42="" ff(l)-5="" b(et)28="" fh(a)917="" fr(1)957="" b(:)g(:)g(:)f(;)h(a)1249="" fq(n)1323="" ff(and)28="" fh(b)1580="" fr(1)1619="" b(:)g(:)g(:)f(;)h(b)1912="" fq(n)1987="" ff(b)-5="" b(e)28="" b(c)-5="" b(ol)5="" b(le)-5="" b(ctions)26="" b(of)i(events)g(in)f(some)g(pr)-5="" b(ob)g(ability)0="" y(sp)g(ac)g(e)34="" b(wher)-5="" b(e:)1299="" fj(pr)1420="" fo(="" )1498="" y(\\)1485="" fq(s)1620="" fh(a)1693="" fq(i)1721="" fo(!)1837="" b(pr)2083="" )2162="" y(\\)2149="" fq(s)2283="" fh(b)2357="" fq(i)2385="" fo(!)0="" ff(for)35="" b(every)f(subset)h="" fh(s)f="" fg(\032)28="" fj([)p="" fj(])35="" ff(such)g(that)g="" fg(j)27="" ff(.)45="" b(then)1010="" fo(\014)1010="" y(\014)1010="" y(\014)1038="" fj(pr)1159="" )1260="" fq(n)1235="" fo([)1224="" fh(a)1428="" fq(i)1457="" fo(!)1544="" fg(\000)23="" fj(pr)1765="" )1866="" fq(n)1842="" fo([)1831="" fh(b)2036="" fq(i)2064="" fo(!)2130="" y(\014)2130="" y(\014)2207="" fh(e)2378="" fr(\012\()2575="" fc(k)2609="" fe(2)p="" fc(n)11="" fe(log)h="" fc(n)2708="" fr(\))0="" ff(mor)-5="" b(e)g(over,)34="" b(ther)-5="" b(e)35="" b(ar)-5="" b(o)g(e\016cients)34="" fh(\025)1421="" fq(j)1485="" fh(\025)1645="" fq(j;k)r(;n)1833="" ff(such)35="" b(that)811="" fo(\014)811="" y(\014)811="" y(\014)880="" fq(k)840="" fo(x)838="" fq(j)t="" fh(\025)1035="" fq(j)1114="" fo(x)1088="" fq(j)1277="" fj(pr)1398="" )1477="" y(\\)1464="" fq(s)1598="" fh(a)1671="" fq(i)1700="" fo(!)1787="" fj(pr)2008="" )2109="" fq(n)2085="" fo([)2074="" fh(a)2278="" fq(i)2306="" fo(!)2372="" y(\014)2372="" y(\014)2427="" fh(e)2577="" fr(\012\()2775="" fc(k)2809="" fc(n)10="" fe(log)i="" fc(n)2908="" ff(and)34="" b(these)h(c)-5="" fh(\025)990="" fq(j)1061="" ff(c)-5="" b(an)34="" b(b)-5="" b(found)f(in)h(time)g(p)-5="" b(olynomial)33="" b(in)i="" ff(.)146="" y(on)30="" b(the)h(other)f(hand,)g(families)f="" fh(a)1409="" fr(1)1449="" b(:)g(:)g(:)f(;)h(a)1741="" fq(n)1818="" ff(and)30="" fh(b)2077="" fr(1)2116="" b(:)g(:)g(:)f(;)h(b)2409="" fq(n)2486="" ff(exist)30="" b(for)g(which)g="" fj(pr)17="" b(\()3288="" fo(t)3357="" fq(s)3495="" fh(a)3568="" fq(i)3596="" fj(\))40="" y(pr)17="" b(\()159="" fo(t)228="" fq(s)366="" fh(b)440="" fq(i)469="" fj(\))35="" ff(whenever)e="" fh(k)38="" ff(and)992="" fo(\014)992="" y(\014)992="" y(\014)1020="" fj(pr)1141="" )1242="" fq(n)1217="" fo([)1206="" fh(a)1410="" fq(i)1438="" fo(!)1526="" fj(pr)1747="" )1848="" fq(n)1824="" fo([)1813="" fh(b)2017="" fq(i)2046="" fo(!)2111="" y(\014)2111="" y(\014)2189="" fg(\025)50="" fh(e)2361="" fq(o)r="" fr(\()2509="" fc(k)2543="" fe(2)2588="" y(log)12="" fc(n)p="" a(n)2726="" fd(remark)37="" b(2.2)o(:)44="" fj(w)-8="" b(e)33="" b(follo)m(w)d(the)i(notation)f(in)g([4])h="" (and)g(denote)h(b)m(y)g="" fh(e)6="" fh(k)s(;)17="" b(n)p="" fj(\))32="" b(the)g(maxim)m(um)e(di\013er-)0="" y(ence)38="" b(b)s(et)m(w)m(een)h(pr\()p="" fh(a)883="" fq(i)912="" fj(\))e(and)g(pr\()p="" fh(b)1463="" fq(i)1492="" fj(\),)g(where)i(there)e(are)g="" fh(n)g="" fj(ev)m(en)m(ts)i="" fh(a)2769="" fq(i)2834="" fj(and)e="" fh(b)3497="" fq(i)3562="" fj(suc)m(h)0="" y(that)34="" b(the)g(in)m(tersection)g="" (of)f(an)m(y)i="" fg(\024)30="" fh(k)37="" fj(of)c(the)h="" fh(a)1754="" fq(i)1783="" fj('s)g(has)g(the)g(same)g(probabilit)m(y)e(of)="" h(the)i(corresp)s(ond-)0="" y(ing)43="" b(ev)m(en)m(t)j(with)e="" fh(b)751="" fq(i)780="" fj('s.)79="" b(the)45="" b(theorem)f(states)h="" (that)f="" fj(\))48="" fg(\000)2826="" fj(~)2812="" y(\002\()2936="" fq(k)2975="" fq(n)3019="" fj(\)\).)78="" b(our)45="" b(b)s(elief)d(is)0="" y(that)31="" b(the)i(logarithmic)28="" b(terms)j(hidden)h(in)f(the)h="" (\\soft)g(\002")f(notation)f(are)i(redundan)m(t)h(and)f(the)g(truth)0="" y(is)38="" fj(\))39="" fj(\002\()955="" fq(k)994="" fq(n)1038="" fj(\)\).)63="" b(moreo)m(v)m(er,)42="" b(w)m(e)e(think)f(that)f(the)i(presen)="" m(t)g(metho)s(ds)f(are)g(p)s(o)m(w)m(erful)0="" y(enough)33="" b(to)f(establish)g(this)h(statemen)m(t)g(and)f(no)h(essen)m(tial)f(new)="" i(ideas)e(will)f(b)s(e)h(required.)p="" a(com)m(bining)f(t)m(w)m(o)i(lemmas)e(from)g([4])i(\(pp.)44="" b(354,)31="" b(357\))h(w)m(e)i(get)e(the)h(follo)m(wing:)1851="" y(3)p="" fd(lemma)37="" b(2.3)9="" b(:)66="" fj(\))46="" fg(\024)h="" fh(\016)i="" ff(i\013)44="" b(e)45="" b(is)g(a)g(r)-5="" b(e)g(al)44="" b(p)-5="" b(olynomial)44="" fh(q)49="" ff(of)c(de)-5="" b(gr)g(e)g(e)44="" b(at)h(most)g="" fh(k)j="" ff(whose)0="" y(c)-5="" b(onstant)43="" b(term)g(is)f(zer)-5="" b(o,)45="" b(so)d(that)i="" fh(q)t="" fg(\000)i="" fg(j)42="" b(\024)1999="" fq(\016)p="" fr(2)p="" fq(\016)2131="" ff(holds)g(for)h(every)g(inte)-5="" b(ger)42="" fh(m)h="" b(:)g(:)g(:)e(;)i(n)p="" ff(.)0="" y(mor)-5="" b(if)h="" fh(x)p="" fj(\))28="" fo(p)950="" fq(k)950="" y(j)t="" fh(\025)1150="" fq(j)1187="" fo(\020)1236="" fq(x)1240="" y(j)1276="" fo(\021)1326="" ff(,)34="" b(then)980="" fo(\014)980="" y(\014)980="" y(\014)1050="" fq(k)1009="" fo(x)1008="" fh(\025)1204="" fq(j)1283="" fo(x)1257="" fq(j)1447="" fj(pr)1568="" )1646="" y(\\)1633="" fq(s)1768="" fh(a)1841="" fq(i)1869="" fo(!)1957="" fg(\000)22="" fj(pr)2177="" )2279="" fq(n)2254="" fo([)2243="" fh(a)2447="" fq(i)2475="" fo(!)2541="" y(\014)2541="" y(\014)2596="" fh(\016)n(:)0="" fd(pro)s(of:)70="" fj(\(of)45="" b(theorem)h(2.1\))f(w)-8="" b(e)47="" b(follo)m(w)d(the)i="" (approac)m(h)g(suggested)h(b)m(y)g(lemma)d(2.3)i(and)g(ex-)0="" y(plicitly)c(construct)j(p)s(olynomials)c="" fj(\))k(that)f(satisfy)g(the)h(lemma)d(with)i="" fh(\016)51="" fj(\012\()3500="" fq(k)3539="" fq(n)11="" fr(log)i="" fq(n)3647="" fj(\)\).)0="" y(the)41="" b(co)s(e\016cien)m(ts)h(of)e(this)g(p)s(olynomial,)f="" (expressed)k(as)e(a)f(linear)f(com)m(bination)f(of)3191="" fo(\020)3241="" fq(x)3245="" y(j)3281="" fo(\021)3371="" fj(o)m(v)m(er)j="" fh(j)47="" y(1)p="" b(:)g(:)g(:)f(;)h(k)45="" fj(will)40="" b(satisfy)i(the)g(claim)e(made)i(in)g(the)g(theorem.)73="" b(by)43="" b(a)f(simple)f(c)m(hange)i(of)f(v)-5="" b(ariable,)0="" y(w)m(e)51="" b(need)h(to)e(construct)h(a)f(real)g(p)s(olynomial)d="" fh(t)j="" fj(of)g(degree)h="" fj(whic)m(h)d(satis\014es)g="" fh(t)p="" fj(\))58="" b(and)0="" y(\001)28="" fg(\025)g="" fj(max)396="" fq(m)p="" fq(;:::)o(;n)p="" fr(1)800="" fj(,)k(where)i(\001)28="" fj(\012\()2052="" fq(k)2091="" fq(n)2199="" fj(\)\).)146="" y(t)-8="" b(o)41="" b(b)s(egin,)i(w)m(e)f="" (c)m(ho)s(ose)g(for)f="" fh(t)g="" fj(to)g(v)-5="" b(anish)41="" b(at)f(in)m(teger)h(p)s(oin)m(ts)g(near)g(the)h(ends)g(of)f(the)g(in)m="" (terv)-5="" b(al)0="" y([0)p="" b(:)g(:)g(:)f(;)h(n)22="" fg(\000)g="" fj(1].)44="" b(that)33="" b(is,)f(w)m(e)h(let)1262="" fh(s)p="" fq(a)p="" fr(1)1581="" fo(y)1616="" fr(0)1698="" fh(x)22="" fg(\000)h="" fh(i)p="" fg(\001)2056="" fr(1)2070="" fo(y)2059="" fq(b)2189="" fh(x)h="" fg(\000)f="" fh(j)6="" y(and)45="" fj(\))j(:=")g" fj(\).)79="" b(in)m(tegers)g="" fh(a)f="" fj(and)h="" fh(b)g="" fj(dep)s(end)g(on)g="" fh(n)f="" fj(and)h(their)f(sum)g(is)g="" fj(.)79="" y(maxim)m(um)30="" b(of)j="" fg(j)f="" fj(o)m(v)m(er)i="" fh(m)28="" fj(1)32="" b(is)g(the)i(maxim)m(um)c(of)i="" fg(j)2677="" fq(s)p="" fr(\))p="" fr(\))2837="" fg(j)g="" fj(o)m(v)m(er)i(the)f(same)g(set)g(of)0="" fj('s.)44="" b(a)33="" b(direct)f(calculation)f(with)h(the)h(p)s="" (olynomial)c="" fh(s)j="" fj(yields:)244="" fg(j)282="" fj(\))499="" fo(\020)717="" fq(m)730="" y(a)779="" fo(\021\020)879="" fr(1)989="" fq(b)1129="" fo(\021)p="" a(\020)902="" fq(n)904="" y(a)944="" fo(\021)1216="" fg(\024)1331="" fo(\020)1421="" fq(n)1381="" y(a)p="" fr(+)p="" fq(b)1503="" a(\020)1421="" fq(n)1423="" y(a)1464="" fo(\021)1591="" fj(2)1745="" fq(h)5="" fr(\()1915="" fc(a)p="" fe(+)p="" fc(b)p="" a(n)2033="" fq(h)g="" fr(\()2218="" a(n)2264="" fr(\)+)p="" fq(o)p="" fr(\(1\)\))2502="" fh(:)1096="" fj(\(1\))146="" y(if)35="" fh(k)g="" fg(\025)d="" fj(3)p="" fh(n=")p" fj(5)j(w)m(e)h(select)f="" fh(a)d="" fj(2.)51="" b(it)35="" b(is)f(easily)g(v)m(eri\014ed)i(that)f(for)f(ev)m(ery)="" j(c)m(hoice)e(of)g="" fh(m)g="" fj(in)f(our)0="" y(range,)f(the)g(righ)m(t)="" e(hand)i(side)g(in)f(equation)g(1)g(is)g(exp)s(onen)m(tially)g(small)e="" (in)i="" fj(,)h(as)g(needed.)146="" y(f)-8="" b(or)29="" fh(k)i(<)d="" fj(5)h(a)h(more)f(complicated)f="" (construction)i(is)g(called)f(for.)42="" b(w)-8="" b(e)30="" b(still)e(guaran)m="" (tee)i(that)g="" fh(t)0="" fj(v)-5="" b(anishes)30="" b(on)f(in)m(tegral)e="" (p)s(oin)m(ts)i(near)g(the)h(ends)g(of)f(the)g(in)m(terv)-5="" b(al)28="" b([0)p="" b(:)g(:)g(:)f(;)h(n)e="" fj(1].)43="" b(around)29="" b(the)h(cen)m(ter)0="" y(of)43="" b(this)f(in)m(terv)-5="" b(al,)45="" b(w)m(e)f(con)m(trol)e(the)h(gro)m(wth)h="" (of)e="" fh(t)i="" fj(using)e(a)h(\(linearly)e(transformed\))i(tc)m(heb)m="" (yshef)0="" y(p)s(olynomial.)d(let)1120="" fr(1)1439="" fo(y)1475="" fr(0)1556="" fh(x)23="" fj(\))h="" fg(\001)1914="" fr(1)1928="" fo(y)1917="" fq(b)2047="" fh(x)g="" fg(\001)f="" fh(\034)2460="" fq(r)2499="" y(where)34="" fh(\034)324="" fq(r)395="" fj(is)e(a)g(linearly)f(transformed)h(tc)m(heb)m="" (yshef)k(p)s(olynomial:)1397="" fh(\034)1439="" fq(r)1478="" fh(t)1797="" fq(r)1835="" fj(\()1966="" fh(x)c="" fh(a)p="" a(n)h="" fh(b)h="" fh(a)2288="" fj(here)36="" fh(t)290="" fq(r)328="" fj(\))g(is)f(the)h="" fh(r)s="" fj(-th)e(tc)m(heb)m(yshef)39="" b(p)s(olynomial,)33="" b(and)j="" fh(a)24="" fj(+)g="" fh(r)36="" fj(.)52="" b(e)36="" b(also)e(let)h="" fh(\013)e="" fh(a="n)0" fj(and)i="" fh(\014)i="" fh(b="n)p" fj(.)50="" b(or)34="" fh(n)24="" fh(b)32="" fh(m)f="" fg(\025)h="" fh(a)j="" fj(the)g(same)g(calculation)d="" (carried)j(out)f(for)g="" fh(k)h="" fj(5)i(can)h(b)s(e)1851="" y(4)p="" fj(rep)s(eated.)56="" b(since)37="" b(a)g(tc)m(heb)m(yshef)j="" (p)s(olynomial)33="" b(v)-5="" b(aries)36="" b(b)s(et)m(w)m(een)j="" fj(1)d(and)h(1)f(when)i(the)f(argumen)m(t)0="" y(is)32="" b(a)g(real)g(in)g([)p="" fj(1],)32="" b(w)m(e)i(conclude)f(that)244="" fj(2)708="" fq(\013)p="" fq(\014)s="" fr(\)\))1317="" b(t)1450="" fq(r)1488="" fj(\()1618="" fh(n)22="" a(n)g="" fh(a)h="" fh(b)1941="" fh(:)1619="" fj(\(2\))146="" y(the)23="" b(tc)m(heb)m(yshef)="" i(p)s(olynomial)18="" b(can)k(b)s(e)g(written)g(as)g="" fh(t)2115="" fq(r)2153="" fr(1)p="" a(2)2487="" fo(\020)2537="" fj(+)2750="" fh(x)2888="" fr(2)2950="" fj(1\))3137="" fq(r)3197="" fj(+)f(\()p="" fg(\000)3510="" fh(x)3648="" fr(2)3710="" fj(1)o(\))3896="" fq(r)3934="" fo(\021)3984="" fj(,)0="" y(whic)m(h)33="" b(is)f(con)m(v)m(enien)m(t)j(for)d(estimations.)41="" b(using)33="" b(this)f(expression,)i(it)d(not)i(hard)f(to)h(sho)m(w)g(that:)847="" fh(t)904="" fq(r)942="" fj(\()1072="" fh(b)1395="" fh(t)1621="" fq(r)1659="" fj(\()1798="" y(1)22="" fh(\013)p="" fj(1)g="" fh(\013)h="" fh(\014)2132="" fj(\(1)22="" b(+)2510="" fo(q)p="" fj(2)p="" fh(\014)5="" fj(\))2740="" fq(r)2778="" fj(t)-8="" b(o)31="" b(get)g(an)g(upp)s(er)g(b)s(ound)h(on)f="" fg(j)1186="" fr(\))1345="" fj(,)g(w)m(e)h(select)g="" fh(\013)c="" fq(k)p="" a(n)12="" fr(log)h="" fq(n)2366="" fj(\))30="" b(and)h="" fh(\014)j="" fh(\013)2877="" fr(2)2916="" fj(.)43="" b(ogether)32="" b(with)f(the)0="" y(lo)m(w)m(er)i(b)s(ound)f(on)h="" fh(t)748="" fq(r)786="" fj(,)g(w)m(e)g(obtain,)f(after)g(some)h="" (calculations,)d(an)j(upp)s(er)g(b)s(ound)g(in)f(equation)g(2:)941="" fg(8)p="" fh(m)d="" b(:)g(:)g(:)f(;)h(n)110="" fg(j)1687="" fj(\))1904="" fj(exp)q(\()p="" fj(\012\()2538="" fh(k)2592="" fh(n)17="" fj(log)g="" fh(n)2733="" fj(\)\))0="" y(as)33="" b(claimed.)146="" b(e)38="" b(no)m(w)f(turn)g(to)g="" (a)g(lo)m(w)m(er)g(b)s(ound)g(on)g="" fj(\).)56="" b(by)38="" b(a)e(sligh)m(t)g(mo)s(di\014cation)e(of)j="" (lemma)e(2.3,)0="" y(this)28="" b(amoun)m(ts)g(to)h(sho)m(wing)f(that)g="" (there)i(is)e(no)g(p)s(olynomial)d="" fh(t)k="" fj(of)f(degree)h="" fj(with)d="" fj(\(0\))g(=")g(1)g(and)g(with)0" fh(<)h(\017)i="" fj(for)g(ev)m(ery)h="" b(:)g(:)g(:)f(;)h(n)29="" fj(where)j="" fh(\017)c="" fj(\012\()2257="" fq(k)2296="" fe(2)2343="" fr(log)13="" a(n)2499="" fj(\)\).)42="" b(letting)29="" fj(\))f(=")g(1)16" b(+)3440="" fo(p)3528="" fq(k)3528="" fr(1)3587="" fh(a)3638="" fq(i)3666="" fh(x)3721="" fq(i)0="" fj(w)m(e)28="" b(need)f(to)g(sho)m(w)g(that)g(the)g(follo)="" m(wing)d(system)k(of)e(linear)f(inequalities)g(\(in)g(the)i="" fh(a)3035="" fq(i)3064="" fj(\))f(is)h(inconsisten)m(t:)1027="" b(:)g(:)g(:)e(;)i(n)133="" fh(\017)28="" b(<)f="" fj(1)22="" b(+)2237="" fq(k)2196="" fo(x)2199="" fh(a)2384="" fq(i)2412="" fh(m)2497="" fq(i)2553="" fh(<)28="" b(\017:)0="" fj(inconsistency)38="" b(will)33="" b(b)s(e)i(established)h(b)m(y)g(linearly)e(com)m(bining)g="" fh(k)27="" fj(+)d(1)35="" b(of)g(these)i(inequalities.)50="" b(our)0="" y(plan)33="" b(is)g(to)g(\014nd)h="" fh(x)688="" fr(1)728="" b(:)g(:)g(:)e(;)i(x)1001="" fq(k)r="" fr(+1)1168="" fj(\(in)m(tegers\))34="" b(whic)m(h)g(are)f(the)h(indices)f(for)g(inequalities)f(participating)0="" y(in)27="" b(this)g(com)m(bination,)g(the)h="" fh(x)1091="" fq(j)1128="" fj(-th)f(inequalit)m(y)g(b)s(eing)g(w)m(eighed)="" h(b)m(y)g="" fh(w)2552="" fq(j)2616="" b(:)g(:)g(:)f(;)h(k)e="" fj(+)d(1\).)41="" b(\(in)28="" b(fact,)0="" y(lp)i(dualit)m(y)f(sa)m(ys)i(w)m(e)g(ha)m(v)m="" (e)g(no)f(c)m(hoice)g(here,)h(and)f(this)f(is)g="" ff(the)h="" fj(w)m(a)m(y)h(to)f(deriv)m(e)g(a)g(con)m(tradiction\).)41="" b(a)0="" y(con)m(tradiction)31="" b(is)i(obtained)f(if)f(follo)m(wing)f="" (conditions)h(hold:)1279="" fh(i)d="" b(:)g(:)g(:)e(;)i(k)1948="" fr(+1)1952="" fo(x)1951="" fh(w)2163="" fq(j)2199="" fh(x)2254="" fq(i)2254="" y(j)2319="" b(0)0="" b(means)g(that)f(all)f="" (nonconstan)m(t)i(terms)g(get)f(eliminated,)e(and)1502="" fr(+1)1507="" fo(x)1549="" fr(1)1647="" fh(w)1717="" fq(j)1781="">)e(\017)1941 4269 y Fq(k)r Fr(+1)1945 4294 y Fo(X)1988 4475 y Fr(1)2086 4377 y Fg(j)p Fh(w)2184 4392 y Fq(j)2220 4377 y Fg(j)0 4634 y Fj(whic)m(h)33 b(means)f(that)g(the)h(com)m(bination)d(of)i (constan)m(t)h(terms)g(is)f(a)g(con)m(tradiction.)42 b(It)32 b(is)g(con)m(v)m(enien)m(t)0 4755 y(to)g(normalize)f(with)786 4688 y Fo(P)873 4715 y Fq(k)r Fr(+1)873 4779 y(1)1023 4755 y Fh(w)1093 4770 y Fq(j)1157 4755 y Fj(=)c(1,)33 b(th)m(us)g(transforming)e(the)i(latter)f(condition)f(to:)1618 4959 y(1)p 1618 5004 49 4 v 1623 5095 a Fh(\017)1704 5027 y(>)1808 4919 y Fq(k)r Fr(+1)1812 4944 y Fo(X)1855 5125 y Fr(1)1953 5027 y Fg(j)p Fh(w)2051 5042 y Fq(j)2087 5027 y Fg(j)p Fh(:)1851 5349 y Fj(5)p eop %%Page: 6 6 6 5 bop 0 100 a Fj(Observ)m(e)32 b(that)d(the)h Fh(w)815 115 y Fq(j)881 100 y Fj(satisfy)f(a)h(linear)e(system)i(of)f (equations,)i(and)f(can,)g(therefore)g(b)s(e)g(expressed)0 220 y(in)35 b(terms)h(of)g(the)g Fh(x)733 235 y Fq(j)806 220 y Fj(b)m(y)h(Cramer's)g(rule.)53 b(The)37 b(matrix)e(of)g(this)h (linear)f(system)i(of)e(equations)i(is)e(a)0 340 y(V)-8 b(andermonde,)33 b(so)g(the)g(expressions)h(for)e Fh(w)1660 355 y Fq(j)1729 340 y Fj(are)g(con)m(v)m(enien)m(t:)1410 613 y Fh(w)1480 628 y Fq(j)1544 613 y Fj(=)c Fg(\006)1872 476 y Fo(Q)1950 563 y Fq(i)p Fn(6)p Fr(=)p Fq(j)2082 543 y Fh(x)2137 558 y Fq(i)p 1735 590 568 4 v 1735 615 a Fo(Q)1814 702 y Fq(i)p Fn(6)p Fr(=)p Fq(j)1929 681 y Fj(\()p Fh(x)2022 696 y Fq(i)2073 681 y Fg(\000)23 b Fh(x)2228 696 y Fq(j)2265 681 y Fj(\))2312 613 y Fh(:)0 892 y Fj(Our)33 b(goal)d(is,)j(then,)g(to)f(\014nd)h(in)m(tegers)g Fh(x)1511 907 y Fr(1)1551 892 y Fh(;)17 b(:)g(:)g(:)f(;)h(x)1825 907 y Fq(k)r Fr(+1)1990 892 y Fj(for)32 b(whic)m(h)1430 1076 y Fq(k)r Fr(+1)1435 1101 y Fo(X)1433 1283 y Fq(j)t Fr(=1)1576 1184 y Fg(j)1622 1101 y Fo(Y)1620 1286 y Fq(i)p Fn(6)p Fr(=)p Fq(j)1865 1117 y Fh(x)1920 1132 y Fq(i)p 1758 1161 298 4 v 1758 1252 a Fh(x)1813 1267 y Fq(i)1863 1252 y Fg(\000)23 b Fh(x)2018 1267 y Fq(j)2065 1184 y Fg(j)k Fh(<)2234 1117 y Fj(1)p 2234 1161 49 4 v 2239 1252 a Fh(\017)2293 1184 y(:)0 1509 y Fj(Our)37 b(c)m(hoice)g(for)f (the)i Fh(x)880 1524 y Fq(i)945 1509 y Fj(is)f(as)g(follo)m(ws:)51 b(Let)37 b Fh(R)f Fj(:=)f Fg(b)2020 1470 y Fq(k)2059 1447 y Fe(2)p 2020 1486 74 4 v 2035 1544 a Fq(n)2103 1509 y Fg(c)p Fj(.)57 b(F)-8 b(or)36 b Fh(i)f Fj(=)g(1)p Fh(;)17 b(:)g(:)g(:)f(;)h(R)37 b Fj(w)m(e)h(let)f Fh(x)3317 1524 y Fq(i)3380 1509 y Fj(=)e Fh(i)p Fj(.)57 b(F)-8 b(or)0 1647 y Fh(R)29 b(<)e(i)h Fg(\024)g Fh(k)e Fj(+)c(1)32 b(w)m(e)i(let)d Fh(x)967 1662 y Fq(i)1024 1647 y Fj(=)c Fg(b)1181 1608 y Fq(i)1205 1584 y Fe(2)p 1182 1624 59 4 v 1185 1681 a Fq(R)1250 1647 y Fg(c)p Fj(.)0 1925 y Fd(Prop)s(osition)36 b(2.4)o(:)45 b Ff(With)35 b(the)g(ab)-5 b(ove)34 b(choic)-5 b(e)34 b(of)g(the)h Fh(x)2119 1940 y Fq(i)2148 1925 y Ff(,)g(for)f(every)h Fh(i)p Ff(,)1213 2080 y Fo(Y)1210 2264 y Fq(j)t Fn(6)p Fr(=)p Fq(i)1479 2095 y Fh(x)1534 2110 y Fq(j)p 1348 2139 353 4 v 1348 2231 a Fg(j)p Fh(x)1431 2246 y Fq(i)1482 2231 y Fg(\000)22 b Fh(x)1636 2246 y Fq(j)1673 2231 y Fg(j)1738 2163 y Fh(<)28 b Fj(exp)q(\()p Fh(O)s Fj(\()p Fh(R)17 b Fj(log)f Fh(n)p Fj(\)\))p Fh(:)0 2521 y Fd(Remark)37 b(2.5)o(:)45 b Fj(F)-8 b(or)32 b(future)h(reference,)h(let)e Fh(Y)54 b Fj(b)s(e)32 b(the)h(left)f(expression)i(in)e(this)g(inequalit)m(y)-8 b(.)p 3523 2529 34 67 v 0 2799 a Fd(Pro)s(of:)42 b Fj(Calculate)30 b(the)h(n)m(umerator)g(\014rst.)43 b(Whether)32 b Fh(x)2111 2814 y Fq(i)2170 2799 y Fj(is)f(missing)e(from)g(this)i(pro)s(duct)g (is)f(incon-)0 2919 y(sequen)m(tial)j(for)f(the)h(t)m(yp)s(e)g(of)f (estimate)g(w)m(e)i(are)e(seeking.)892 3126 y Fo(Y)1015 3209 y Fh(x)1070 3224 y Fq(j)1134 3209 y Fj(=)c Fh(R)q Fj(!)1408 3101 y Fq(k)r Fr(+1)1419 3126 y Fo(Y)1357 3309 y Fq(j)t Fr(=)p Fq(R)p Fr(+1)1588 3209 y Fg(b)1642 3141 y Fh(j)1688 3105 y Fr(2)p 1642 3185 86 4 v 1647 3277 a Fh(R)1737 3209 y Fg(c)g(\024)g Fh(R)q Fj(!)2033 3126 y Fo(Y)2166 3141 y Fh(j)2212 3105 y Fr(2)p 2166 3185 V 2171 3277 a Fh(R)2289 3209 y Fj(=)2429 3141 y(\()p Fh(k)d Fj(+)d(1\)!)2755 3105 y Fr(2)p 2402 3185 419 4 v 2402 3277 a Fh(R)q Fj(!)p Fh(R)2579 3248 y Fq(k)r Fn(\000)p Fq(R)p Fr(+1)2831 3209 y Fh(:)0 3508 y Fj(No)m(w)k(let)f(us)h(turn)g (to)f(the)h(denominator.)40 b(Here)26 b(w)m(e)h(ha)m(v)m(e)g(to)e (distinguish)f(b)s(et)m(w)m(een)k(the)e(cases)h(where)0 3628 y Fh(i)33 b Fj(is)f(among)f(the)i(\014rst)g Fh(R)h Fj(indices,)e(or)g(is)h(bigger.)146 3798 y(If)g(1)27 b Fg(\024)h Fh(i)g Fg(\024)h Fh(R)k Fj(and)g Fh(x)944 3813 y Fq(i)1000 3798 y Fj(=)28 b Fh(i)p Fj(,)33 b(then)533 4005 y Fo(Y)432 4189 y Fr(1)p Fn(\024)p Fq(j)t Fn(6)p Fr(=)p Fq(i)p Fn(\024)p Fq(R)758 4088 y Fg(j)p Fh(i)22 b Fg(\000)h Fh(x)996 4103 y Fq(j)1033 4088 y Fg(j)f(\001)1184 3980 y Fq(k)r Fr(+1)1195 4005 y Fo(Y)1132 4189 y Fq(j)t Fr(=)p Fq(R)p Fr(+1)1363 4088 y Fj(\()p Fh(x)1456 4103 y Fq(j)1515 4088 y Fg(\000)h Fh(i)p Fj(\))28 b Fg(\025)g Fj(\()p Fh(i)22 b Fg(\000)h Fj(1\)!\()p Fh(R)g Fg(\000)g Fh(i)p Fj(\)!)2527 3980 y Fq(k)r Fr(+1)2538 4005 y Fo(Y)2476 4189 y Fq(j)t Fr(=)p Fq(R)p Fr(+1)2707 4088 y Fj(\()2755 4021 y Fh(j)2801 3984 y Fr(2)p 2755 4065 86 4 v 2760 4156 a Fh(R)2872 4088 y Fg(\000)g Fh(i)f Fg(\000)h Fj(1\))k Fg(\025)0 4481 y Fj(\()p Fh(i)19 b Fg(\000)g Fj(1\)!\()p Fh(R)g Fg(\000)g Fh(i)p Fj(\)!)650 4373 y Fq(k)r Fr(+1)661 4398 y Fo(Y)642 4582 y Fq(R)p Fr(+2)813 4414 y Fj(\()p Fh(j)28 b Fg(\000)22 b Fh(R)h Fg(\000)g Fj(1\)\()p Fh(j)28 b Fj(+)22 b 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134 a Fq(R)2204 100 y Fg(c)p Fj(.)44 b(In)33 b(this)f(case,)307 293 y Fq(R)281 317 y Fo(Y)273 500 y Fq(j)t Fr(=1)395 400 y Fj(\()p Fh(x)488 415 y Fq(i)539 400 y Fg(\000)23 b Fh(j)6 b Fj(\))933 317 y Fo(Y)739 502 y Fq(R)p Fr(+1)p Fn(\024)p Fq(j)t Fn(6)p Fr(=)p Fq(i)p Fn(\024)p Fq(k)r Fr(+1)1249 400 y Fg(j)p Fh(x)1332 415 y Fq(i)1383 400 y Fg(\000)22 b Fh(x)1537 415 y Fq(j)1574 400 y Fg(j)28 b(\025)1843 330 y(b)1897 290 y Fq(i)1921 267 y Fe(2)p 1898 306 V 1901 364 a Fq(R)1966 330 y Fg(c)p Fj(!)p 1745 377 392 4 v 1745 477 a Fg(b)1799 438 y Fq(i)1823 419 y Fe(2)p 1799 454 59 4 v 1802 512 a Fq(R)1890 477 y Fg(\000)22 b Fh(R)q Fg(c)p Fj(!)2168 400 y Fg(\001)2478 317 y Fo(Y)2218 502 y Fq(R)p Fr(+1)p Fn(\024)p Fq(j)t Fn(6)p Fr(=)p Fq(i;i)p Fr(+1)p Fn(\024)p Fq(k)r Fr(+1)2872 333 y Fg(j)p Fh(j)2946 297 y Fr(2)3007 333 y Fg(\000)g Fh(i)3139 297 y Fr(2)3201 333 y Fg(\000)h Fh(i)p Fg(j)p 2872 377 491 4 v 3079 469 a Fh(R)3400 400 y Fg(\025)622 646 y Fo( )698 725 y Fh(i)731 689 y Fr(2)793 725 y 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