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On Intrinsic Bounds in the Nullstellensatz

T. Krick, J. Sabia, P. Solernó
<span title="1997-01-27">1997</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lnpibodwo5cpnp3qbzbnlmdepu" style="color: black;">Applicable Algebra in Engineering, Communication and Computing</a> </i> &nbsp;
In this paper we exhibit a new effective Nullstellensatz which doesn't depend so much on the degree of the involved polynomials as the ones mentioned above, but on a more intrinsic invariant: the geometric  ...  In this sense, our effective Nullstellensatz can be considered more intrinsic and improves the known ones (see Example 3 of Section 4).  ...  Let us consider the classic example: Example 2 In this simple example one easily sees that the maximum of the degrees of the involved polynomials must occur in the stated upper bounds: Here f , . . . ,  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s002000050057">doi:10.1007/s002000050057</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mabl5l5khbf3lhgy5sodo27q3e">fatcat:mabl5l5khbf3lhgy5sodo27q3e</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170809100124/http://mate.dm.uba.ar/~psolerno/krick-sabia-solerno.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/9b/10/9b1050f26503d816c145671946e96147d13971c7.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s002000050057"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Precise sequential and parallel complexity bounds for quantifier elimination over algebraically closed fields

Noaï Fitchas, André Galligo, Jacques Morgenstern
<span title="">1990</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/74pjbehpfraq5ogj3loetdwp7m" style="color: black;">Journal of Pure and Applied Algebra</a> </i> &nbsp;
Due to recent progress concerning Triviality Testing of Polynomial Ideals (relying on effective affine Nullstellensatze) we are able to give upper bounds in a refined and satisfactory precise form.  ...  The new outcomes concern parallelism where the number of processors is controlled by the intrinsic sequential complexity of quantifier elimination.  ...  [7, S] ) with the methods of [19] , one obtains the same precise sequential complexity bounds as in [lo] and [ 181.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0022-4049(90)90159-f">doi:10.1016/0022-4049(90)90159-f</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/uqufovnhp5dwfhxk7ir6mk6opa">fatcat:uqufovnhp5dwfhxk7ir6mk6opa</a> </span>
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Page 5507 of Mathematical Reviews Vol. , Issue 2002H [page]

<span title="">2002</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The search for effective versions of the Nullstellensatz has a long history, and similar but less sharp estimates were known before. The first degree bound was given by G. Hermann in 1926 [Math.  ...  Section four is devoted to the proof of a refined “intrinsic” ver- sion of the main theorem where the estimates on the right-hand sides depend on more complicated measures of geometric and  ... 
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Page 8485 of Mathematical Reviews Vol. , Issue 2000m [page]

<span title="">2000</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
(RA-UNLPS; La Plata) On the intrinsic complexity of the arithmetic Nullstellensatz. (English summary) J. Pure Appl. Algebra 146 (2000), no. 2, 103-183.  ...  The complexity bounds are expressed in terms of the geometric degree and height of the system, which are intrinsic parameters For the web version of Mathematical Reviews, see http: //www.ams.org/mathscinet  ... 
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Page 7270 of Mathematical Reviews Vol. , Issue 2004i [page]

<span title="">2004</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
This paper examines the effect of homogenization on the polyno- mial degree of refutations in the polynomial calculus (PC) and the Hilbert Nullstellensatz (HN) proof systems.  ...  O(1) result is to give a lower bound on the degree of homogenized PC refutations and thus using the previously mentioned result get a lower bound HN degree.  ... 
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Bounds for the Hubert function of polynomial ideals and for the degrees in the Nullstellensatz

Martin Sombra
<span title="">1997</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/74pjbehpfraq5ogj3loetdwp7m" style="color: black;">Journal of Pure and Applied Algebra</a> </i> &nbsp;
We present a new effective Nullstellensatz with bounds for the degrees which depend not only on the number of variables and on the degrees of the input polynomials but also on an additional parameter called  ...  The obtained bound is polynomial in these parameters. It is essentially optimal in the general case, and it substantially improves the existent bounds in some special cases.  ...  The bounds so obtained are more intrinsic and refined than the usual estimates, and we show that they are sharper in some special cases.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0022-4049(97)00028-5">doi:10.1016/s0022-4049(97)00028-5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/uu3t7ue4tncgdgs5otae4nogxa">fatcat:uu3t7ue4tncgdgs5otae4nogxa</a> </span>
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Strong majorization in a free ✱-algebra

J. William. Helton, Scott McCullough, Mihai Putinar
<span title="2006-08-23">2006</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2kqurccfrzb6rdh2cfrpx2gcmu" style="color: black;">Mathematische Zeitschrift</a> </i> &nbsp;
We study, in the spirit of modern real algebra, the interplay between left ideals of the free * -algebra F with n generators, and their suitably defined zero sets; and similarly between quadratic submodules  ...  An Approximate Nullstellensatz on Arbitrary Varieties In this section we propose a Nullstellensatz for an arbitrary non-commutative polynomial q vanishing on a basic algebraic set in F.  ...  In this section the results demand consideration of tuples of bounded operators X on a possibly infinite dimensional space rather than only tuples of matrices.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00209-006-0032-0">doi:10.1007/s00209-006-0032-0</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cwczk3zibjdq7i7d4xvt3pz2we">fatcat:cwczk3zibjdq7i7d4xvt3pz2we</a> </span>
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Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz [article]

Martin Sombra
<span title="1996-10-04">1996</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We present a new effective Nullstellensatz with bounds for the degrees which depend not only on the number of variables and on the degrees of the input polynomials but also on an additional parameter called  ...  The obtained bound is polynomial in these parameters. It is essentially optimal in the general case, and it substantially improves the existent bounds in some special cases.  ...  The Effective Nullstellensatz and the Representation Problem in Complete Intersections In this section we consider the problem of bounding the degrees of the polynomials in the Nullstellensatz and in the  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/alg-geom/9610006v1">arXiv:alg-geom/9610006v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zidh5zjvwzbydafirzqirggspu">fatcat:zidh5zjvwzbydafirzqirggspu</a> </span>
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Sharp estimates for the arithmetic Nullstellensatz [article]

Teresa Krick, Luis Miguel Pardo, Martin Sombra
<span title="1999-11-14">1999</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We also present degree and height estimates of intrinsic type, which depend mainly on the degree and the height of the input polynomial system.  ...  We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over . The result improves previous work of Philippon, Berenstein-Yger and Krick-Pardo.  ...  In Chapter 4, we focus on the intrinsic and sparse versions of the arithmetic Nullstellensatz.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/9911094v1">arXiv:math/9911094v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gzpsu7s7w5hpxe55lsgrypslou">fatcat:gzpsu7s7w5hpxe55lsgrypslou</a> </span>
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Equations for the projective closure and effective Nullstellensatz

L. Caniglia, A. Galligo, J. Heintz
<span title="">1991</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
The proof of the two algorithms are based on an effective Nullstellensatz. Let k be a field.  ...  Heintz, Equations for the projective closure and effective Nullstellensatz, Discrete Applied Mathematics 33 (1991) 1 l-23.  ...  The proof of these two algorithms are based on an effective Nullstellensatz (see [5, 10,11, 20, 13, 8] ) and on a consequence of Bezout's inequality (see [19] ) detailed in Section 1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0166-218x(91)90105-6">doi:10.1016/0166-218x(91)90105-6</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qpprukcd35cnhn6cccsubc7wua">fatcat:qpprukcd35cnhn6cccsubc7wua</a> </span>
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Recent improvements in the complexity of the effective Nullstellensatz

Carlos A. Berenstein, Daniele C. Struppa
<span title="">1991</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wsx3rzhpingfvewcn5nwhfkq3e" style="color: black;">Linear Algebra and its Applications</a> </i> &nbsp;
We bring up to date the estimates on the complexity of the effective Nullstellensatz and the membership problem.  ...  The authors are grateful to J. Heintz and B. Sh@inan fw providing them with preprints of their works.  ...  These works (see e. ideals are intrinsically more complex (in the sense of complexity theory) than in the case of homogeneous ideals.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0024-3795(91)90115-d">doi:10.1016/0024-3795(91)90115-d</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/poiopjbuwzbddjgfxubpgdwm6i">fatcat:poiopjbuwzbddjgfxubpgdwm6i</a> </span>
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Sharp estimates for the arithmetic Nullstellensatz

Mart�n Sombra, Luis Miguel Pardo, Teresa Krick
<span title="">2001</span> <i title="Duke University Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qiuuwifskbf4foqk2k7hj4ilce" style="color: black;">Duke mathematical journal</a> </i> &nbsp;
We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integer ring Z Z .  ...  We also present degree and height estimates of intrinsic type, which depend mainly on the degree and the height of the input polynomial system.  ...  In Chapter 4, we focus on the intrinsic and sparse versions of the arithmetic Nullstellensatz.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1215/s0012-7094-01-10934-4">doi:10.1215/s0012-7094-01-10934-4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/w4fwucpczvamrmadhejntumfta">fatcat:w4fwucpczvamrmadhejntumfta</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20031231085302/http://maply.univ-lyon1.fr:80/~sombra/papers/arith-nss/maino.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d1/76/d176f09d09601ceb808e080868714105b74026a9.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1215/s0012-7094-01-10934-4"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Sharp estimates for the arithmetic Nullstellensatz

Mart�n Sombra, Luis Miguel Pardo, Teresa Krick
<span title="">2001</span> <i title="Duke University Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qiuuwifskbf4foqk2k7hj4ilce" style="color: black;">Duke mathematical journal</a> </i> &nbsp;
We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integer ring Z Z .  ...  We also present degree and height estimates of intrinsic type, which depend mainly on the degree and the height of the input polynomial system.  ...  In Chapter 4, we focus on the intrinsic and sparse versions of the arithmetic Nullstellensatz.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1215/dmj/1000314065">doi:10.1215/dmj/1000314065</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6oecluyftvd5bb4uqzjcbkpdia">fatcat:6oecluyftvd5bb4uqzjcbkpdia</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20061002171449/http://mate.dm.uba.ar/~krick/KrPaSo00.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f0/df/f0dfac9038cc01c83c95d0e554beb22028dc1eec.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1215/dmj/1000314065"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Page 4168 of Mathematical Reviews Vol. , Issue 98G [page]

<span title="">1998</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
On intrinsic bounds in the Nullstellensatz. (English summary) Appl. Algebra Engrg. Comm.  ...  In this article the authors produce an up- per bound for this D, which in some cases improves the existing bounds.  ... 
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Heights of varieties in multiprojective spaces and arithmetic Nullstellensätze

Carlos D'Andrea, Teresa Krick, Martín Sombra
<span title="">2013</span> <i title="Societe Mathematique de France"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ca7ifai33je6xc4hduaf346wpe" style="color: black;">Annales Scientifiques de l&#39;Ecole Normale Supérieure</a> </i> &nbsp;
We present bounds for the degree and the height of the polynomials arising in some problems in effective algebraic geometry including the implicitization of rational maps and the effective Nullstellensatz  ...  Our treatment is based on arithmetic intersection theory in products of projective spaces and extends to the arithmetic setting constructions and results due to Jelonek.  ...  Besides of their intrinsic interest, these results play a central role in our treatment of the parametric and arithmetic Nullstellensätze.  ... 
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