Relevant bibliographies by topics / Polynomial calculus

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Polynomial calculus'

Author: Grafiati
Published: 4 June 2021
Last updated: 11 February 2022

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Journal articles on the topic "Polynomial calculus"

1

Bulmer, M., D. Fearnley-Sander, and T. Stokes. "Towards a calculus of algorithms." Bulletin of the Australian Mathematical Society 50, no. 1 (August 1994): 81–89. http://dx.doi.org/10.1017/s000497270000959x.

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We develop a generalised polynomial formalism which captures the concept of an algebra of piece-wise denned polynomials. The formalism is based on the Boolean power construction of universal algebra. A generalisation of the theory of substitution homomorphisms is developed. The abstract operation of composition of generalised polynomials in one variable is denned and shown to correspond to function composition.
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FUKUHARA, SHINJI, YUKIO MATSUMOTO, and NORIKO YUI. "NON-COMMUTATIVE POLYNOMIAL RECIPROCITY FORMULAE." International Journal of Mathematics 12, no. 08 (November 2001): 973–86. http://dx.doi.org/10.1142/s0129167x01001088.

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We prove non-commutative reciprocity formulae for certain polynomials using Fox's free differential calculus. The abelianizations of these reciprocity formulae rediscover the polynomial reciprocity formulae of Carlitz and Berndt–Dieter. Further, many other reciprocity formulae related to Dedekind sums are rederived from our polynomial reciprocity formulae; these include, for instance, generalizations of the classical Eisenstein reciprocity formula.
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Songhafouo Tsopméné, Paul Arnaud, and Donald Stanley. "Polynomial functors in manifold calculus." Topology and its Applications 248 (October 2018): 75–116. http://dx.doi.org/10.1016/j.topol.2018.08.012.

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Filmus, Yuval, Massimo Lauria, Jakob Nordström, Noga Ron-Zewi, and Neil Thapen. "Space Complexity in Polynomial Calculus." SIAM Journal on Computing 44, no. 4 (January 2015): 1119–53. http://dx.doi.org/10.1137/120895950.

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Buresh-Oppenheim, Joshua, Matthew Clegg, Russell Impagliazzo, and Toniann Pitassi. "Homogenization and the polynomial calculus." Computational Complexity 11, no. 3-4 (June 1, 2002): 91–108. http://dx.doi.org/10.1007/s00037-002-0171-6.

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Stick, Marvin E. "Maclaurin Taylor Series for Transcendental Functions: A Graphing-Calculator View of Convergence." Mathematics Teacher 92, no. 9 (December 1999): 833–37. http://dx.doi.org/10.5951/mt.92.9.0833.

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Most calculus students can perform the manipulation necessary for a polynomial approximation of a transcendental function. However, many do not understand the underlying concept. Graphing-calculator technology can be used to bridge this gap between the concept of an interval of convergence for a series and polynomial approximations. Calculus-reform textbooks usually treat this topic by displaying lower-order Maclaurin series approximations to selected transcendental functions to encourage discussions of intervals of convergence. Some textbooks display lower-order Taylor polynomials for ln x expanded about x = 1. This article presents a way to examine the topic in more depth.
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Gatto, Letterio, and Taíse Santiago. "Schubert Calculus on a Grassmann Algebra." Canadian Mathematical Bulletin 52, no. 2 (June 1, 2009): 200–212. http://dx.doi.org/10.4153/cmb-2009-023-x.

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AbstractThe (classical, small quantum, equivariant) cohomology ring of the grassmannian G(k, n) is generated by certain derivations operating on an exterior algebra of a free module of rank n (Schubert calculus on a Grassmann algebra). Our main result gives, in a unified way, a presentation of all such cohomology rings in terms of generators and relations. Using results of Laksov and Thorup, it also provides a presentation of the universal factorization algebra of a monic polynomial of degree n into the product of two monic polynomials, one of degree k.
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Pavlyuk, A. M. "HOMFLY Polynomial Invariants of Torus Knots and Bosonic (q, p)-Calculus." Ukrainian Journal of Physics 58, no. 12 (December 2013): 1178–81. http://dx.doi.org/10.15407/ujpe58.12.1178.

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Grant, Melva R., William Crombie, Mary Enderson, and Nell Cobb. "Polynomial calculus: rethinking the role of calculus in high schools." International Journal of Mathematical Education in Science and Technology 47, no. 6 (January 12, 2016): 823–36. http://dx.doi.org/10.1080/0020739x.2015.1133851.

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Galesi, Nicola, and Massimo Lauria. "On the Automatizability of Polynomial Calculus." Theory of Computing Systems 47, no. 2 (February 14, 2009): 491–506. http://dx.doi.org/10.1007/s00224-009-9195-5.

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Dissertations / Theses on the topic "Polynomial calculus"

1

Mikša, Mladen. "On Complexity Measures in Polynomial Calculus." Doctoral thesis, KTH, Teoretisk datalogi, TCS, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-197278.

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Proof complexity is the study of different resources that a proof needs in different proof systems for propositional logic. This line of inquiry relates to the fundamental questions in theoretical computer science, as lower bounds on proof size for an arbitrary proof system would separate P from NP. We study two simple proof systems: resolution and polynomial calculus. In resolution we reason using clauses, while in polynomial calculus we use polynomials. We study three measures of complexity of proofs: size, space, and width/degree. Size is the number of clauses or monomials that appear in a resolution or polynomial calculus proof, respectively. Space is the maximum number of clauses/monomials we need to keep at each time step of the proof. Width/degree is the size of the largest clause/monomial in a proof. Width is a lower bound for space in resolution. The original proof of this claim used finite model theory. In this thesis we give a different, more direct proof of the space-width relation. We can ask whether a similar relation holds between space and degree in polynomial calculus. We make some progress on this front by showing that when a formula F requires resolution width w then the XORified version of F requires polynomial calculus space Ω(w). We also show that space lower bounds do not imply degree lower bounds in polynomial calculus. Width/degree and size are also related, as strong lower bounds for width/degree imply strong lower bounds for size. Currently, proving width lower bounds has a well-developed machinery behind it. However, the degree measure is much less well-understood. We provide a unified framework for almost all previous degree lower bounds. We also prove some new degree and size lower bounds. In addition, we explore the relation between theory and practice by running experiments on some current state-of-the-art SAT solvers.

QC 20161206


Understanding the Hardness of Theorem Proving
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2

Curzi, Gianluca. "Non-laziness in implicit computational complexity and probabilistic λ-calculus." Thesis, Université de Paris (2019-....), 2020. http://www.theses.fr/2020UNIP7010.

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Cette thèse explore les avantages de la “non-paresse” dans la Complexité Computationnelle implicite et dans le lambda calcul probabiliste. Plus précisément, cette thèse peut être divisée en deux parties principales. Dans la première, nous étudions dans tous les sens les mécanismes d’effacement et de duplication linéaire, qui nous conduisent aux systèmes de typage LEM (Lin- early Exponential Multiplicative Type Assignment) et LAM (Linearly Additive Multiplicative Type Assignment). Le premier est capable d’exprimer des versions plus faibles des règles exponentielles de la Logique Linéaire, tandis que le second est doté de règles additifs plus faibles, appelées additifs linéaires. Ces systèmes bénéficient respectivement d’une élimination de coupure cubique et d’un résultat de normalisation linéaire. Étant donné que les additifs linéaires ne nécessitent pas d’évaluation paresseuse pour éviter l’explosion exponentielle de la normalisation (contraire- ment aux additifs standard), ils peuvent être utilisés pour obtenir une caractérisation implicite des fonctions calculables en temps polynomial probabiliste qui ne dépend pas du choix de la stratégie de réduction. Ce résultat a été réalisé dans STA, un système obtenu en dotant STA (Soft Type Assignment) d’une formulation aléatoire des additifs linéaires, qui capture aussi les classes de complexité PP et BPP. La deuxième partie de la thèse est centrée sur le lambda calcul probabiliste doté d’une sémantique opérationnelle basée sur la réduction de tête, c’est-à-dire une politique d’évaluation non paresseuse d’appel par nom. Nous prouvons que la bisimilarité probabiliste est “fully abstract” par rapport à l’équivalence observationnelle. Ce résultat témoigne le pouvoir discriminant de la non-paresse, qui permet de retrouver une correspondance parfaite entre les deux équivalences qui manquait dans le cadre paresseux. De plus, nous montrons que la similarité probabiliste est correcte mais pas complète par rapport au préordre observationnel
This thesis explores the benefits of non-laziness in both Implicit Computational Complexity and probabilistic computation. More specifically, this thesis can be divided in two main parts. In the first one, we investigate in all directions the mechanisms of linear erasure and duplication, which lead us to the type assignment systems LEM (Linearly Exponential Multiplicative Type Assignment) and LAM (Linearly Additive Multiplicative Type Assignment). The former is able to express weaker versions of the exponential rules of Linear Logic, while the latter has weaker additive rules, called linear additives. These systems enjoy, respectively, a cubic cut-elimination and a linear normalization result. Since linear additives do not require a lazy evaluation to avoid the exponential blow up in normalization (unlike the standard additives), they can be employed to obtain an implicit characterization of the functions computable in probabilistic polynomial time that does not depend on the choice of the reduction strategy. This result is achieved in STA⊕, a system that extends STA (Soft Type Assignment) with a randomized formulation of linear additives. Also, this system is able to capture the complexity classes PP and BPP. The second part of the thesis is focused on the probabilistic λ-calculus endowed with an operational semantics based on the head reduction, i.e. a non-lazy call-by-name evaluation policy. We prove that probabilistic applicative bisimilarity is fully abstract with respect to context equivalence. This result witnesses the discriminating power of non-laziness, which allows to recover a perfect match between the two equivalences that was missing in the lazy setting. Moreover, we show that probabilistic applicative similarity is sound but not complete for the context preorder
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Araaya, Tsehaye. "The Symmetric Meixner-Pollaczek polynomials." Doctoral thesis, Uppsala University, Department of Mathematics, 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3501.

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The Symmetric Meixner-Pollaczek polynomials are considered. We denote these polynomials in this thesis by pn(λ)(x) instead of the standard notation pn(λ) (x/2, π/2), where λ > 0. The limiting case of these sequences of polynomials pn(0) (x) =limλ→0 pn(λ)(x), is obtained, and is shown to be an orthogonal sequence in the strip, S = {z ∈ ℂ : −1≤ℭ (z)≤1}.

From the point of view of Umbral Calculus, this sequence has a special property that makes it unique in the Symmetric Meixner-Pollaczek class of polynomials: it is of convolution type. A convolution type sequence of polynomials has a unique associated operator called a delta operator. Such an operator is found for pn(0) (x), and its integral representation is developed. A convolution type sequence of polynomials may have associated Sheffer sequences of polynomials. The set of associated Sheffer sequences of the sequence pn(0)(x) is obtained, and is found

to be ℙ = {{pn(λ) (x)} =0 : λ ∈ R}. The major properties of these sequences of polynomials are studied.

The polynomials {pn(λ) (x)}∞n=0, λ < 0, are not orthogonal polynomials on the real line with respect to any positive real measure for failing to satisfy Favard’s three term recurrence relation condition. For every λ ≤ 0, an associated nonstandard inner product is defined with respect to which pn(λ)(x) is orthogonal.

Finally, the connection and linearization problems for the Symmetric Meixner-Pollaczek polynomials are solved. In solving the connection problem the convolution property of the polynomials is exploited, which in turn helps to solve the general linearization problem.

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Heinz, Sebastian. "Preservation of quasiconvexity and quasimonotonicity in polynomial approximation of variational problems." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2008. http://dx.doi.org/10.18452/15808.

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Die vorliegende Arbeit beschäftigt sich mit drei Klassen ausgewählter nichtlinearer Probleme, die Forschungsgegenstand der angewandten Mathematik sind. Diese Probleme behandeln die Minimierung von Integralen in der Variationsrechnung (Kapitel 3), das Lösen partieller Differentialgleichungen (Kapitel 4) und das Lösen nichtlinearer Optimierungsaufgaben (Kapitel 5). Mit deren Hilfe lassen sich unterschiedlichste Phänomene der Natur- und Ingenieurwissenschaften sowie der Ökonomie mathematisch modellieren. Als konkretes Beispiel werden mathematische Modelle der Theorie elastischer Festkörper betrachtet. Das Ziel der vorliegenden Arbeit besteht darin, ein gegebenes nichtlineares Problem durch polynomiale Probleme zu approximieren. Um dieses Ziel zu erreichen, beschäftigt sich ein großer Teil der vorliegenden Arbeit mit der polynomialen Approximation von nichtlinearen Funktionen. Den Ausgangspunkt dafür bildet der Weierstraßsche Approximationssatz. Auf der Basis dieses bekannten Satzes und eigener Sätze wird als Hauptresultat der vorliegenden Arbeit gezeigt, dass im Übergang von einer gegebenen Funktion zum approximierenden Polynom wesentliche Eigenschaften der gegebenen Funktion erhalten werden können. Die wichtigsten Eigenschaften, für die dies bisher nicht bekannt war, sind: Quasikonvexität im Sinne der Variationsrechnung, Quasimonotonie im Zusammenhang mit partiellen Differentialgleichungen sowie Quasikonvexität im Sinne der nichtlinearen Optimierung (Theoreme 3.16, 4.10 und 5.5). Schließlich wird gezeigt, dass die zu den untersuchten Klassen gehörenden nichtlinearen Probleme durch polynomiale Probleme approximiert werden können (Theoreme 3.26, 4.16 und 5.8). Die dieser Approximation zugrunde liegende Konvergenz garantiert sowohl eine Approximation im Parameterraum als auch eine Approximation im Lösungsraum. Für letztere werden die Konzepte der Gamma-Konvergenz (Epi-Konvergenz) und der G-Konvergenz verwendet.
In this thesis, we are concerned with three classes of non-linear problems that appear naturally in various fields of science, engineering and economics. In order to cover many different applications, we study problems in the calculus of variation (Chapter 3), partial differential equations (Chapter 4) as well as non-linear programming problems (Chapter 5). As an example of possible applications, we consider models of non-linear elasticity theory. The aim of this thesis is to approximate a given non-linear problem by polynomial problems. In order to achieve the desired polynomial approximation of problems, a large part of this thesis is dedicated to the polynomial approximation of non-linear functions. The Weierstraß approximation theorem forms the starting point. Based on this well-known theorem, we prove theorems that eventually lead to our main result: A given non-linear function can be approximated by polynomials so that essential properties of the function are preserved. This result is new for three properties that are important in the context of the considered non-linear problems. These properties are: quasiconvexity in the sense of the calculus of variation, quasimonotonicity in the context of partial differential equations and quasiconvexity in the sense of non-linear programming (Theorems 3.16, 4.10 and 5.5). Finally, we show the following: Every non-linear problem that belongs to one of the three considered classes of problems can be approximated by polynomial problems (Theorems 3.26, 4.16 and 5.8). The underlying convergence guarantees both the approximation in the parameter space and the approximation in the solution space. In this context, we use the concepts of Gamma-convergence (epi-convergence) and of G-convergence.
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Souvay, Arnaud. "Une approche intrinsèque des foncteurs de Weil." Thesis, Université de Lorraine, 2012. http://www.theses.fr/2012LORR0257/document.

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Nous construisons un foncteur de la catégorie des variétés sur un corps ou un anneau topologique K, de caractéristique arbitraire, dans la catégorie des variétés sur A, où A est une algèbre de Weil, c'est-à-dire une K-algèbre de la forme A = K + N, où N est un idéal nilpotent. Le foncteur correspondant, noté T^A, et appelé foncteur de Weil, peut être interprété comme un foncteur d'extension scalaire de K à A. Il est construit à l'aide des polynômes de Taylor, dont nous donnons une définition en caractéristique quelconque. Ce résultat généralise à la fois des résultats connus pour les variétés réelles ordinaires, et les résultats obtenus dans le cas des foncteurs tangents itérés et dans le cas des anneaux de jets (A = K[X]/(X^{k+1})). Nous montrons que pour toute variété M, T^A M possède une structure de fibré polynomial sur M, et nous considérons certains aspects algébriques des foncteurs de Weil, notamment ceux liés à l'action du « groupe de Galois » Aut_K(A). Nous étudions les connexions, qui sont un outil important d'analyse des fibrés, dans deux contextes différents : d'une part sur les fibrés T^A M, et d?autre part sur des fibrés généraux sur M, en suivant l'approche d'Ehresmann. Les opérateurs de courbure d'une connexion sont induits par l'action du groupe de Galois Aut_K(A) et ils forment une obstruction à l'« intégrabilité » d'une connexion K-lisse en une connexion A-lisse
We construct a functor from the category of manifolds over a general topological base field or ring K, of arbitrary characteristic, to the category of manifolds over A, where A is a so-called Weil algebra, i.e. a K-algebra of the form A = K + N, where N is a nilpotent ideal. The corresponding functor, denoted by T^A, and called a Weil functor, can be interpreted as a functor of scalar extension from K to A. It is constructed by using Taylor polynomials, which we define in arbitrary characteristic. This result generalizes simultaneously results known for ordinary, real manifolds, and results for iterated tangent functors and for jet rings (A = K[X]/(X^{k+1})). We show that for any manifold M, T^A M is a polynomial bundle over M, and we investigate some algebraic aspects of the Weil functors, in particular those related to the action of the "Galois group" Aut_K(A). We study connections, which are an important tool for the analysis of fiber bundles, in two different contexts : connections on the Weil bundles T^A M, and connections on general bundles over M, following Ehresmann's approach. The curvature operators are induced by the action of the Galois group Aut_K(A) and they form an obstruction to the "integrability" of a K-smooth connection to an A-smooth one
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Schimanski, Stefan. "Polynomial Time Calculi." Diss., lmu, 2009. http://nbn-resolving.de/urn:nbn:de:bvb:19-99100.

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Vinyals, Marc. "Space in Proof Complexity." Doctoral thesis, KTH, Teoretisk datalogi, TCS, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-206571.

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ropositional proof complexity is the study of the resources that are needed to prove formulas in propositional logic. In this thesis we are concerned with the size and space of proofs, and in particular with the latter. Different approaches to reasoning are captured by corresponding proof systems. The simplest and most well studied proof system is resolution, and we try to get our understanding of other proof systems closer to that of resolution. In resolution we can prove a space lower bound just by showing that any proof must have a large clause. We prove a similar relation between resolution width and polynomial calculus space that lets us derive space lower bounds, and we use it to separate degree and space. For cutting planes we show length-space trade-offs. This is, there are formulas that have a proof in small space and a proof in small length, but there is no proof that can optimize both measures at the same time. We introduce a new measure of space, cumulative space, that accounts for the space used throughout a proof rather than only its maximum. This is exploratory work, but we can also prove new results for the usual space measure. We define a new proof system that aims to capture the power of current SAT solvers, and we show a landscape of length-space trade-offs comparable to those in resolution. To prove these results we build and use tools from other areas of computational complexity. One area is pebble games, very simple computational models that are useful for modelling space. In addition to results with applications to proof complexity, we show that pebble game cost is PSPACE-hard to approximate. Another area is communication complexity, the study of the amount of communication that is needed to solve a problem when its description is shared by multiple parties. We prove a simulation theorem that relates the query complexity of a function with the communication complexity of a composed function.

QC 20170509

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Verron, Thibaut. "Régularisation du calcul de bases de Gröbner pour des systèmes avec poids et déterminantiels, et application en imagerie médicale." Thesis, Paris 6, 2016. http://www.theses.fr/2016PA066355/document.

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La résolution de systèmes polynomiaux est un problème aux multiples applications, et les bases de Gröbner sont un outil important dans ce cadre. Il est connu que de nombreux systèmes issus d'applications présentent une structure supplémentaire par rapport à des systèmes arbitraires, et que ces structures peuvent souvent être exploitées pour faciliter le calcul de bases de Gröbner.Dans cette thèse, on s'intéresse à deux exemples de telles structures, pour différentes applications. Tout d'abord, on étudie les systèmes homogènes avec poids, qui sont homogènes si on calcule le degré en affectant un poids à chaque variable. Cette structure apparaît naturellement dans de nombreuses applications, dont un problème de cryptographie (logarithme discret). On montre comment les algorithmes existants, efficaces pour les polynômes homogènes, peuvent être adaptés au cas avec poids, avec des bornes de complexité générique divisées par un facteur polynomial en le produit des poids.Par ailleurs, on étudie un problème de classification de racines réelles pour des variétés définies par des déterminants. Ce problème a une application directe en théorie du contrôle, pour l'optimisation de contraste de l'imagerie à résonance magnétique. Ce système particulier s'avère insoluble avec les stratégies générales pour la classification. On montre comment ces stratégies peuvent tirer profit de la structure déterminantielle du système, et on illustre ce procédé en apportant des réponses aux questions posées par le problème d'optimisation de contraste
Polynomial system solving is a problem with numerous applications, and Gröbner bases are an important tool in this context. Previous studies have shown that systèmes arising in applications usually exhibit more structure than arbitrary systems, and that these structures can be used to make computing Gröbner bases easier.In this thesis, we consider two examples of such structures. First, we study weighted homogeneous systems, which are homogeneous if we give to each variable an arbitrary degree. This structure appears naturally in many applications, including a cryptographical problem (discrete logarithm). We show how existing algorithms, which are efficient for homogeneous systems, can be adapted to a weighted setting, and generically, we show that their complexity bounds can be divided by a factor polynomial in the product of the weights.Then we consider a real roots classification problem for varieties defined by determinants. This problem has a direct application in control theory, for contrast optimization in magnetic resonance imagery. This specific system appears to be out of reach of existing algorithms. We show how these algorithms can benefit from the determinantal structure of the system, and as an illustration, we answer the questions from the application to contrast optimization
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Monnot, Jérôme. "Familles d'instances critiques et approximation polynomiale." Paris 9, 1998. https://portail.bu.dauphine.fr/fileviewer/index.php?doc=1998PA090028.

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S'il est facile de concevoir que le comportement d'un algorithme approché diffère suivant l'instance à traiter, quel est réellement le rôle de ces instances dans l'approximation d'un problème ? Quelles influences ont-elles sur le comportement d'un algorithme ? A quel niveau ? Pour répondre à ces questions, nous avons construit un formalisme permettant d'ordonner les instances selon le niveau d'information qu'elles procurent. Ce formalisme nous conduit à définir des familles d'instances dépendant soit d'un problème soit d'un triplet (problème, algorithme approche, mesure d'approximation). La classification en familles critiques et fortement critiques permet de comprendre, d'analyser et d'éventuellement améliorer le comportement de cet algorithme au pire des cas. Afin de construire de telles familles, nous proposons divers outils et méthodes basés sur la notion centrale de plus mauvaise instance. D'un point de vue opérationnel, concernant la mesure différentielle, nous avons spécifié certaines formes d'instances empêchant la très bonne approximabilité et caractérise les classes ptas() et fptas(). Ensuite, nous avons obtenu, grâce aux familles critiques, des résultats d'approximation positifs et négatifs sur une classe de problèmes de partitionnement définis par certaines propriétés logiques. Les problèmes de la coloration, du bin-packing, de l'ensemble dominant de Steiner en font partie. Précisons toutefois qu'une étude spécifique de ces problèmes met en évidence des différences d'approximation : la coloration appartient à apx()ptas() tandis que le bin-packing appartient à ptas()isfptas(). Enfin, par les mêmes procédés, nous obtenons de bons résultats d'approximation pour les problèmes connexes à celui de l'arbre de Steiner
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Wailly, Olivier. "Placement optimal de capteurs sur un système à modèle polynomial." Corte, 2004. http://www.theses.fr/2005CORT3091.

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La présente thèse met en évidence une méthode novatrice dans le placement de capteurs sur un système automatisé. Cette méthode n'est applicable que sur les systèmes présentant un modèle polynomial. Cette méthode est innovante dans l'utilisation du calcul formel, et plus spécifiquement des algorithmes des bases de Groëbner. Après avoir montré comment il est possible, utile et interressant d'utiliser le calcul formel, il est développé une recherche de placement prenant en compte des critères optimums. Ainsi, des algorithmes avec critères de coût et critères de fiabilité sont élaborés
The present thesus present a novel method in sensor design on automated system. This method is only applicable on polynomial systems. This method is using symbolic calculus software. Especially, the Groënberg bases' algorithms are used. After showing the interest of this method, algoritms and programs with optimal criteria are presented. So, the criteria like cost and reliability are developed
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Books on the topic "Polynomial calculus"

1

F, Pixley Alden, ed. Polynomial completeness in algebraic systems. Boca Raton, Fla: Chapman & Hall/CRC, 2001.

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Pitt, François. The bounded linear calculus: A characterization of the class of polynomial-time computable functions based on bounded linear logic. Toronto: University of Toronto, Dept. of Computer Science, 1994.

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Uncommon mathematical excursions: Polynomia and related realms. Washington, D.C: Mathematical Association of America, 2009.

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From polynomials to sums of squares. Bristol: Institute of Physics Pub., 1995.

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5

R, Ziegler Michael, Byleen Karl, Barnett Raymond A, and Barnett Raymond A, eds. Additional calculus topics: To accompany Calculus, 10/e, and College mathematics, 10/e. Upper Saddle River, N.J: Prentice Hall, 2006.

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R, Ziegler Michael, and Byleen Karl E, eds. Additional calculus topics: To accompany Calculus, 11/e, and College mathematics, 11/e. Upper Saddle River, N.J: Pearson, 2007.

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7

Bucchianico, Alessandro Di. Probabilistic and analytical aspects of the umbral calculus. [Amsterdam, Netherlands]: Centrum voor Wiskunde en Informatica, 1997.

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Real solutions to equations from geometry. Providence, R.I: American Mathematical Society, 2011.

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International Conference on p-Adic Functional Analysis (11th 2010 Université Blaise Pascal). Advances in non-Archimedean analysis: Eleventh International Conference on p-Adic Functional Analysis, July 5-9 2010, Université Blaise Pascal, Clermont-Ferrand, France. Edited by Araujo-Gomez Jesus 1965-, Diarra B. (Bertin) 1944-, and Escassut Alain. Providence, R.I: American Mathematical Society, 2011.

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PISRS 2011 International Conference on Analysis, Fractal Geometry, Dynamical Systems and Economics (2011 Messina, Italy). Fractal geometry and dynamical systems in pure and applied mathematics. Edited by Carfi David 1971-, Lapidus, Michel L. (Michel Laurent), 1956-, Pearse, Erin P. J., 1975-, Van Frankenhuysen Machiel 1967-, and Mandelbrot Benoit B. Providence, Rhode Island: American Mathematical Society, 2013.

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Book chapters on the topic "Polynomial calculus"

1

Packel, Ed, and Stan Wagon. "Polynomial Approximation and Taylor Series." In Animating Calculus, 247–59. New York, NY: Springer New York, 1997. http://dx.doi.org/10.1007/978-1-4612-2408-2_21.

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Bonacina, Ilario. "Space in Polynomial Calculus." In Space in Weak Propositional Proof Systems, 41–57. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-73453-8_4.

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Buresh-Oppenheim, Josh, Toniann Pitassi, Matt Clegg, and Russell Impagliazzo. "Homogenization and the Polynomial Calculus." In Automata, Languages and Programming, 926–38. Berlin, Heidelberg: Springer Berlin Heidelberg, 2000. http://dx.doi.org/10.1007/3-540-45022-x_78.

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López, César Pérez. "Polynomial Divisibility, Interpolation, and Algebraic Extensions." In MATLAB Symbolic Algebra and Calculus Tools, 73–132. Berkeley, CA: Apress, 2014. http://dx.doi.org/10.1007/978-1-4842-0343-9_3.

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Gaboardi, Marco, and Simona Ronchi Della Rocca. "Type Inference for a Polynomial Lambda Calculus." In Lecture Notes in Computer Science, 136–52. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-02444-3_9.

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Baillot, Patrick, and Virgile Mogbil. "Soft lambda-Calculus: A Language for Polynomial Time Computation." In Lecture Notes in Computer Science, 27–41. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-540-24727-2_4.

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Mitchell, John C. "Probabilistic Polynomial-Time Process Calculus and Security Protocol Analysis." In Programming Languages and Systems, 23–29. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. http://dx.doi.org/10.1007/3-540-45309-1_2.

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de Groote, Philippe. "The Non-associative Lambek Calculus with Product in Polynomial Time." In Lecture Notes in Computer Science, 128–39. Berlin, Heidelberg: Springer Berlin Heidelberg, 1999. http://dx.doi.org/10.1007/3-540-48754-9_14.

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Baillot, Patrick, Erika De Benedetti, and Simona Ronchi Della Rocca. "Characterizing Polynomial and Exponential Complexity Classes in Elementary Lambda-Calculus." In Advanced Information Systems Engineering, 151–63. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-662-44602-7_13.

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Meyer, Roland, Victor Khomenko, and Reiner Hüchting. "A Polynomial Translation of π-Calculus (FCP) to Safe Petri Nets." In Lecture Notes in Computer Science, 440–55. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-32940-1_31.

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Conference papers on the topic "Polynomial calculus"

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Filmus, Yuval, Massimo Lauria, Jakob Nordstrom, Neil Thapen, and Noga Ron-Zewi. "Space Complexity in Polynomial Calculus." In 2012 IEEE Conference on Computational Complexity (CCC). IEEE, 2012. http://dx.doi.org/10.1109/ccc.2012.27.

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Galesi, Nicola, Leszek Kolodziejczyk, and Neil Thapen. "Polynomial Calculus Space and Resolution Width." In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 2019. http://dx.doi.org/10.1109/focs.2019.00081.

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Beck, Chris, Jakob Nordstrom, and Bangsheng Tang. "Some trade-off results for polynomial calculus." In the 45th annual ACM symposium. New York, New York, USA: ACM Press, 2013. http://dx.doi.org/10.1145/2488608.2488711.

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Alekhnovich, M., and A. A. Razborov. "Lower bounds for polynomial calculus: non-binomial case." In Proceedings 42nd IEEE Symposium on Foundations of Computer Science. IEEE, 2001. http://dx.doi.org/10.1109/sfcs.2001.959893.

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Baillot, P., and K. Terui. "Light types for polynomial time computation in lambda-calculus." In Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004. IEEE, 2004. http://dx.doi.org/10.1109/lics.2004.1319621.

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Riis, Søren. "On the Asymptotic Nullstellensatz and Polynomial Calculus Proof Complexity." In 2008 23rd Annual IEEE Symposium on Logic in Computer Science (LICS 2008). IEEE, 2008. http://dx.doi.org/10.1109/lics.2008.30.

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Dokuchaev, V. A., and S. V. Pavlov. "Polynomial approximation of signals as a type of operational calculus." In 2018 Systems of Signal Synchronization, Generating and Processing in Telecommunications (SYNCHROINFO). IEEE, 2018. http://dx.doi.org/10.1109/synchroinfo.2018.8457061.

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Madet, Antoine. "A polynomial time λ-calculus with multithreading and side effects." In the 14th symposium. New York, New York, USA: ACM Press, 2012. http://dx.doi.org/10.1145/2370776.2370785.

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Buss, Sam, Dima Grigoriev, Russell Impagliazzo, and Toniann Pitassi. "Linear gaps between degrees for the polynomial calculus modulo distinct primes." In the thirty-first annual ACM symposium. New York, New York, USA: ACM Press, 1999. http://dx.doi.org/10.1145/301250.301399.

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Hakoniemi, Tuomas. "Monomial size vs. Bit-complexity in Sums-of-Squares and Polynomial Calculus." In 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, 2021. http://dx.doi.org/10.1109/lics52264.2021.9470545.

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