Given two polynomial
s <
span id="mml
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source">P,q
span><
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ss="mathContainer hidden"><
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span> we con
sider the following que
stion: “how large can the index of the fir
st non-zero moment <
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source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002203961500368X&_mathId=si2.gif&_user=111111111&_pii=S002203961500368X&_rdoc=1&_issn=00220396&md5=4fbd410de72af33210ce95b1323cae64">
ss="imgLazyJSB inlineImage" height="26" width="86" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S002203961500368X-si2.gif">script>
style="vertical-align:bottom" width="86" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S002203961500368X-si2.gif">script><
span cla
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span>
span>
span> be, a
ssuming the
sequence i
s not identically zero?” The an
swer
K to thi
s que
stion i
s known a
s the moment Bautin index, and we provide the fir
st general upper bound: <
span id="mml
si3" cla
ss="mathml
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ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S002203961500368X&_mathId=
si3.gif&_u
ser=111111111&_pii=S002203961500368X&_rdoc=1&_i
ssn=00220396&md5=59fa0200e217fff7cc3c00d58c09297c" title="Click to view the MathML
source">K猢?+deg鈦+3(deg鈦&minu
s;1)<
sup>2
sup>
span><
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ss="mathContainer hidden"><
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span>
span>
span>. The proof i
s ba
sed on qualitative analy
si
s of linear ODE
s, applied to Cauchy-type integral
s of certain algebraic function
s.
sp0020">The moment Bautin index plays an important role in the study of bifurcations of periodic solution in the polynomial Abel equation <span id="mmlsi4" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002203961500368X&_mathId=si4.gif&_user=111111111&_pii=S002203961500368X&_rdoc=1&_issn=00220396&md5=0c708b3ff0ade5a81cdcafcf1b080410" title="Click to view the MathML source">y<sup>′sup>=py<sup>2sup>+蔚qy<sup>3sup>span><span class="mathContainer hidden"><span class="mathCode">span>span>span> for <span id="mmlsi5" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002203961500368X&_mathId=si5.gif&_user=111111111&_pii=S002203961500368X&_rdoc=1&_issn=00220396&md5=26be93bfba7296edd5385680b567e44c" title="Click to view the MathML source">p,qspan><span class="mathContainer hidden"><span class="mathCode">span>span>span> polynomials and <span id="mmlsi6" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002203961500368X&_mathId=si6.gif&_user=111111111&_pii=S002203961500368X&_rdoc=1&_issn=00220396&md5=8516f639514b58138c74c43c669dc4b3" title="Click to view the MathML source">蔚鈮?span><span class="mathContainer hidden"><span class="mathCode">span>span>span>. In particular, our result implies that for p satisfying a well-known generic condition, the number of periodic solutions near the zero solution does not exceed <span id="mmlsi7" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002203961500368X&_mathId=si7.gif&_user=111111111&_pii=S002203961500368X&_rdoc=1&_issn=00220396&md5=0fa1e3db2d6f0e102701ce6bc5214377" title="Click to view the MathML source">5+deg鈦+3deg<sup>2sup>鈦span><span class="mathContainer hidden"><span class="mathCode">span>span>span>. This is the first such bound depending solely on the degrees of the Abel equation.