Strictly positive polynomials in the boundary of the SOS cone

We study the boundary of the cone of real polynomials that can be decomposed as a sum of squares (SOS) of real polynomials. This cone is included in the cone of nonnegative polynomials and both cones share a part of their boundary, which corresponds to polynomials that vanish at at least one point. We focus on the part of the boundary which is not shared, corresponding to strictly positive polynomials. For the cases of polynomials of degree 6 in 3 variables and degree 4 in 4 variables, this boundary has been completely characterized by G. Blekherman in a recent work. For the cases of more variables or higher degree, the problem is more complicated and very few examples or general results are known. Assuming a conjecture by D. Eisenbud, M. Green, and J. Harris, we obtain bounds for the maximum number of polynomials that can appear in a SOS decomposition and the maximum rank of the matrices in the Gram spectrahedron. In particular, for the case of homogeneous quartic polynomials in 5 variables, for which the required case of the conjecture has been recently proved, we obtain bounds that improve the general bounds known up to date. Finally, combining theoretical results with computational techniques, we find examples and counterexamples that allow us to better understand which of the results obtained by G. Blekherman can be extended to the general case and show that the bounds predicted by our results are attainable.Comment: 24 page

Symmetric interpolation, Exchange lemma and Sylvester sums.

The theory of symmetric multivariate Lagrange interpolation is a beautiful but rather unknown tool that has many applications. Here we derive from it an Exchange Lemma that allows to explain in a simple and natural way the full description of the double sum expressions introduced by Sylvester in 1853 in terms of subresultants and their Bézout coefficients.Fil: Krick, Teresa Elena Genoveva. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas ; ArgentinaFil: Szanto, Agnes. North Carolina State University; Estados UnidosFil: Valdettaro, Marcelo Alejandro. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Formulas in roots for the subresultants

Los objetos centrales de esta tesis son los polinomios subresultantes de dos polinomios en una variable, que son, en el caso de polinomios con raíces simples, múltiplos escalares de lo que hoy se llama sumas de Sylvester de sus conjuntos de raíces, como demostró J.J.Sylvester en 1853. En primer término presentamos aquí una generalización de las sumas de Sylvester para multiconjuntos de manera que sigue valiendo la relación con los polinomios subresultantes. En el caso en que los multiconjuntos tienen suficientes elementos distintos, esta generalización es particularmente elegante ya que tiene el mismo aspecto que las sumas de Sylvester. Cuando no hay suficientes elementos distintos, nuestra generalización esmás compleja ya que necesita introducir polinomios de Schur. Sin embargo cabe mencionar que ejemplos previos parecen indicar que no se va a poder encontrar ninguna generalización sencilla de las sumas de Sylvester para multiconjuntos arbitrarios. Nuestro enfoque introduce un Lema de intercambio que permite interpolar ciertos polinomios simétricos endistintos conjuntos de nodos. Obtenemos además más aplicaciones naturales de este lema, no sólo a otras propiedades de subresultantes sino también a construcciones relacionadas con matrices de Bézout y bases de Gröbner. Finalmente estudiamos completamente el caso particular de dos polinomios con una sola raíz múltiple cada uno y logramos probar que las subresultantes son, en ese caso, un múltiplo escalar de cierto polinomio de Jacobi, módulo un cambio de variables afín. Esto permite obtener, vía la ecuación diferencial satisfecha por los polinomios de Jacobi, una cota optimal de complejidad para determinar los coeficientes de una subresultante en la base monomial. De este modo logramos mejorar,para esta clase de polinomios, las cotas de complejidad que existen para el cálculo de una subresultante de polinomios arbitrarios.The main objects of this thesis are the subresultant polynomials of two univariate polynomials, which are, for simple-root polynomials, scalar multiples of what is known today as Sylvester sums, as shown by J.J. Sylvester in 1853. First we present a generalization of Sylvester sums for multisets so that the relationship with subresultants still holds. In the case that the multisets have enough different elements, this generalizationis particularly elegant since it has the same shape as Sylvester sums. When there are not enough different elements, our generalization is more complex since it needs to introduce Schur polynomials. However it should be mentioned that previous examples seem to indicate that it will not be possible to obtain any simple generalization of Sylvester sums forarbitrary multisets. Our approach introduces an Exchange lemma which allows to interpolate some symmetric polynomials in different sets of nodes. We also obtain other natural applications of this lemma, not only concerning further properties of subresultants but also other constructions related to Bézout matrices and Gröbner bases. Finally we fully study the particular case of two polynomials with only one multiple root each and provethat their subresultants are scalar multiples of a certain Jacobi polynomial, modulo an affine change of variables. This allows to obtain, using the differential equation satisfied by Jacobi polynomials, an optimal complexity bound for determining the coefficients of a subresultant in the monomial basis. In this way we improve, for this family of polynomials,the existing complexity bounds for computing a subresultant of arbitrary polynomials.Fil: Valdettaro, Marcelo Alejandro. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina

Closed formula for univariate subresultants in multiple roots

We generalize Sylvester single sums to multisets and show that these sums compute subresultants of two univariate polynomials as a function of their roots independently of their multiplicity structure. This is the first closed formula for subresultants in terms of roots that works for arbitrary polynomials, previous efforts only handled special cases. Our extension involves in some cases confluent Schur polynomials and is obtained by using multivariate symmetric interpolation via an Exchange Lemma.Fil: D'Andrea, Carlos. Universidad de Barcelona; EspañaFil: Krick, Teresa Elena Genoveva. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Szanto, Agnes. North Carolina State University; Estados UnidosFil: Valdettaro, Marcelo Alejandro. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Formulas in roots for the subresultants

Los objetos centrales de esta tesis son los polinomios subresultantes de dos polinomios en una variable, que son, en el caso de polinomios con raíces simples, múltiplos escalares de lo que hoy se llama sumas de Sylvester de sus conjuntos de raíces, como demostró J.J.Sylvester en 1853. En primer término presentamos aquí una generalización de las sumas de Sylvester para multiconjuntos de manera que sigue valiendo la relación con los polinomios subresultantes. En el caso en que los multiconjuntos tienen suficientes elementos distintos, esta generalización es particularmente elegante ya que tiene el mismo aspecto que las sumas de Sylvester. Cuando no hay suficientes elementos distintos, nuestra generalización esmás compleja ya que necesita introducir polinomios de Schur. Sin embargo cabe mencionar que ejemplos previos parecen indicar que no se va a poder encontrar ninguna generalización sencilla de las sumas de Sylvester para multiconjuntos arbitrarios. Nuestro enfoque introduce un Lema de intercambio que permite interpolar ciertos polinomios simétricos endistintos conjuntos de nodos. Obtenemos además más aplicaciones naturales de este lema, no sólo a otras propiedades de subresultantes sino también a construcciones relacionadas con matrices de Bézout y bases de Gröbner. Finalmente estudiamos completamente el caso particular de dos polinomios con una sola raíz múltiple cada uno y logramos probar que las subresultantes son, en ese caso, un múltiplo escalar de cierto polinomio de Jacobi, módulo un cambio de variables afín. Esto permite obtener, vía la ecuación diferencial satisfecha por los polinomios de Jacobi, una cota optimal de complejidad para determinar los coeficientes de una subresultante en la base monomial. De este modo logramos mejorar,para esta clase de polinomios, las cotas de complejidad que existen para el cálculo de una subresultante de polinomios arbitrarios.The main objects of this thesis are the subresultant polynomials of two univariate polynomials, which are, for simple-root polynomials, scalar multiples of what is known today as Sylvester sums, as shown by J.J. Sylvester in 1853. First we present a generalization of Sylvester sums for multisets so that the relationship with subresultants still holds. In the case that the multisets have enough different elements, this generalizationis particularly elegant since it has the same shape as Sylvester sums. When there are not enough different elements, our generalization is more complex since it needs to introduce Schur polynomials. However it should be mentioned that previous examples seem to indicate that it will not be possible to obtain any simple generalization of Sylvester sums forarbitrary multisets. Our approach introduces an Exchange lemma which allows to interpolate some symmetric polynomials in different sets of nodes. We also obtain other natural applications of this lemma, not only concerning further properties of subresultants but also other constructions related to Bézout matrices and Gröbner bases. Finally we fully study the particular case of two polynomials with only one multiple root each and provethat their subresultants are scalar multiples of a certain Jacobi polynomial, modulo an affine change of variables. This allows to obtain, using the differential equation satisfied by Jacobi polynomials, an optimal complexity bound for determining the coefficients of a subresultant in the monomial basis. In this way we improve, for this family of polynomials,the existing complexity bounds for computing a subresultant of arbitrary polynomials.Fil: Valdettaro, Marcelo Alejandro. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina

Subresultants of (x−α)m and (x−β)n, Jacobi polynomials and complexity

In an earlier article (Bostan et al., 2017), with Carlos D’Andrea, we described explicit expressions for the coefficients of the order-d polynomial subresultant of (x − α) m and (x − β)n with respect to Bernstein’s set of polynomials {(x − α)j (x − β)d− j , 0 ≤ j ≤ d}, for 0 ≤ d < min{m,n}. The current paper further develops the study of these structured polynomials and shows that the coefficients of the subresultants of (x − α) m and (x − β)n with respect to the monomial basis can be computed in linear arithmetic complexity, which is faster than for arbitrary polynomials. The result is obtained as a consequence of the amazing though seemingly unnoticed fact that these subresultants are scalar multiples of Jacobi polynomials up to an affine change of variables.Fil: Bostan, Alin. Institut National de Recherche en Informatique et en Automatique; FranciaFil: Krick, Teresa Elena Genoveva. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Szanto, Agnes. North Carolina State University; Estados UnidosFil: Valdettaro, Marcelo Alejandro. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Effects of fertilizer type on nitrous oxide emission and ammonia volatilization in wheat and maize crops

About half of the applied nitrogen (N) is not consumed by crops, causing environmental and economic costs. This N can be lost as ammonia (NH3) volatilization, nitrous oxide (N2O) emission or leaching, among others. This work aimed to compare the amount of gaseous N losses using three different fertilizers on two consecutive experiments: one summer crop (maize) and one winter crop (wheat) in the Rolling Pampa, Argentina. The fertilizers used were urea ammonium nitrate (UAN), calcium ammonium nitrate (CAN) and AN+DMPP (ammonium nitrate-based NPK fertilizer with DMPP nitrification inhibitor). NH3 emissions were estimated using a semi open-static absorption system during the first month after fertilization for each experiment. N2O emissions were estimated using vented static chambers during the growing season of each crop. Results show that CAN or AN+DMPP fertilizers used instead of UAN helped to reduce NH3 volatilization by 45–50% and 62–63% on maize and wheat experiments respectively, but failed to reduce N2O emissions. In addition, contrary to the expected, AN+DMPP increased N2O emissions during the maize experiment. The majority of the gaseous N losses occurred at specific moments of the crop cycle (after N fertilization and around leaf senescence). Losses as NH3 volatilization were higher than N2O emissions in the maize experiment, as expected because of the warmer temperature during this summer crop. However, N2O emissions were higher during the wheat crop, emphasizing the importance of factors such as meteorological conditions, previous land-use, residual soil nitrate and stubble quality on the soil.Fil: Vangeli, Sebastián. Instituto Nacional de Tecnología Agropecuaria (INTA). Instituto De Investigación Clima y Agua; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Agronomía. Departamento de Ingeniería Agrícola y Uso de la Tierra. Cátedra de Manejo y Conservación de Suelo; ArgentinaFil: Posse Beaulieu, Gabriela. Instituto Nacional de Tecnología Agropecuaria (INTA). Instituto de Clima y Agua; ArgentinaFil: Beget, Maria Eugenia. Instituto Nacional de Tecnología Agropecuaria (INTA). Instituto de Clima y Agua; Argentina. Universidad de Buenos Aires. Facultad de Agronomía. Departamento de Métodos Cuantitativos y Sistemas de Información; ArgentinaFil: Otero Estrada, Edit. Instituto Nacional de Tecnología Agropecuaria (INTA). Instituto de Suelos; ArgentinaFil: Valdettaro, Roció Antonella. Instituto Nacional de Tecnología Agropecuaria (INTA). Instituto de Suelos; ArgentinaFil: Oricchio, Patricio. Instituto Nacional de Tecnología Agropecuaria (INTA). Instituto de Clima y Agua; ArgentinaFil: Kandus, Patricia. Instituto Nacional de Tecnología Agropecuaria (INTA). Instituto de Genética; ArgentinaFil: Di Bella, Carlos Marcelo. Instituto Nacional de Tecnología Agropecuaria (INTA). Instituto de Clima y Agua; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Agronomía. Departamento de Métodos Cuantitativos y Sistemas de Información; Argentin