Integral polynomials with small discriminants and resultants

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作者：Victor Beresnevich^a ; ^{victor.beresnevich@york.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; ^{vb8@york.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; Vasili Bernik^b ; ^{bernik@im.bas-net.by" class="auth_mail" title="E-mail the corresponding author} ; ^{bernik.vasili@mail.ru" class="auth_mail" title="E-mail the corresponding author} ; Friedrich Gö ; tze^c ; ^{goetze@math.uni-bielefeld.de" class="auth_mail" title="E-mail the corresponding author}
关键词：11J83 ; 11J13 ; 11K60 ; 11K55
刊名：Advances in Mathematics
出版年：2016
出版时间：6 August 2016
年：2016
卷：298
期：Complete
页码：393-412
全文大小：452 K

文摘

Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305102&_mathId=si1.gif&_user=111111111&_pii=S0001870816305102&_rdoc=1&_issn=00018708&md5=b02a93669942fccda168e31ae3a1bb1d" title="Click to view the MathML source">n∈Nclass="mathContainer hidden">class="mathCode">

n \in N

be fixed, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305102&_mathId=si2.gif&_user=111111111&_pii=S0001870816305102&_rdoc=1&_issn=00018708&md5=3ae0d8bfbbdf8f4b05347e4f3ef8a6c4" title="Click to view the MathML source">Q>1class="mathContainer hidden">class="mathCode">

Q > 1

be a real parameter and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305102&_mathId=si3.gif&_user=111111111&_pii=S0001870816305102&_rdoc=1&_issn=00018708&md5=d200cdee0fb98a06eb0dc8e5a823a057" title="Click to view the MathML source">P_n(Q)class="mathContainer hidden">class="mathCode">

P_{n} (Q)

denote the set of polynomials over class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305102&_mathId=si4.gif&_user=111111111&_pii=S0001870816305102&_rdoc=1&_issn=00018708&md5=8cc109ccb5f14c84189a638b9fb90685" title="Click to view the MathML source">Zclass="mathContainer hidden">class="mathCode">

Z

of degree n and height at most Q . In this paper we investigate the following counting problems regarding polynomials with small discriminant class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305102&_mathId=si5.gif&_user=111111111&_pii=S0001870816305102&_rdoc=1&_issn=00018708&md5=ebec00685221dd4d2f8cd9e78ed885b6" title="Click to view the MathML source">D(P)class="mathContainer hidden">class="mathCode">

D (P)

and pairs of polynomials with small resultant class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305102&_mathId=si6.gif&_user=111111111&_pii=S0001870816305102&_rdoc=1&_issn=00018708&md5=5c49d1cb958e0789f80847b9e9100105" title="Click to view the MathML source">R(P₁,P₂)class="mathContainer hidden">class="mathCode">

R (P_{1}, P_{2})

class="label">(i)

given class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305102&_mathId=si20.gif&_user=111111111&_pii=S0001870816305102&_rdoc=1&_issn=00018708&md5=331d6e225db9a5ea2c0faa083f16f519" title="Click to view the MathML source">0≤v≤n−1class="mathContainer hidden">class="mathCode"> $0 \leq v \leq n - 1$ and a sufficiently large Q, estimate the number of polynomials class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305102&_mathId=si8.gif&_user=111111111&_pii=S0001870816305102&_rdoc=1&_issn=00018708&md5=838ca5d6e565edd32723798dbba0cfb7" title="Click to view the MathML source">P∈P_n(Q)class="mathContainer hidden">class="mathCode"> $P \in P_{n} (Q)$ such that

class="formula" id="fm0010">

class="mathml">class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305102&_mathId=si9.gif&_user=111111111&_pii=S0001870816305102&_rdoc=1&_issn=00018708&md5=f0c2ca1abba51e9ad9e7269ac5bf8b4b">class="imgLazyJSB inlineImage" height="19" width="174" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816305102-si9.gif">

titlethe

class="mathContainer hidden">class="mathCode">

0 < | D (P) | \leq Q^{2 n - 2 - 2 v};

class="temp" src="/sd/blank.gif">

class="label">(ii)

given class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305102&_mathId=si10.gif&_user=111111111&_pii=S0001870816305102&_rdoc=1&_issn=00018708&md5=6bdbaa140976c4618e3ae6acdc931463" title="Click to view the MathML source">0≤w≤nclass="mathContainer hidden">class="mathCode"> $0 \leq w \leq n$ and a sufficiently large Q, estimate the number of pairs of polynomials class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305102&_mathId=si11.gif&_user=111111111&_pii=S0001870816305102&_rdoc=1&_issn=00018708&md5=2315931ce053fd414e0f3b7451e364e1" title="Click to view the MathML source">P₁,P₂∈P_n(Q)class="mathContainer hidden">class="mathCode"> $P_{1}, P_{2} \in P_{n} (Q)$ such that

class="formula" id="fm0020">

class="mathml">class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816305102&_mathId=si12.gif&_user=111111111&_pii=S0001870816305102&_rdoc=1&_issn=00018708&md5=c4000d29e3116a52d5d0317bca19b08c" title="Click to view the MathML source">0<|R(P₁,P₂)|≤Q^2n−2w.class="mathContainer hidden">class="mathCode">

0 < | R (P_{1}, P_{2}) | \leq Q^{2 n - 2 w} .

class="temp" src="/sd/blank.gif">

Our main results provide lower bounds within the context of the above problems. We believe that these bounds are best possible as they correspond to the solutions of naturally arising linear optimisation problems. Using a counting result for the number of rational points near planar curves due to R. C. Vaughan and S. Velani we also obtain the complementary optimal upper bound regarding the discriminants of quadratic polynomials.

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