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Analitic and Algebraic Geometry 4

Analitic and Algebraic Geometry 4

**Contributor(s): **Tadeusz Krasiński (Editor), Stanisław Spodzieja (Editor)

**Subject(s): **Philosophy, Logic

**Published by: **Wydawnictwo Uniwersytetu Łódzkiego

**Keywords: **algebraic Geometry; Analitic Geometry

**Summary/Abstract: **This volume (the fourth in the series) is dedicated to two mathematicians: Wojciech Kucharz, who celebrates 70th anniversary in 2022 and Tadeusz Winiarski who celebrated the 80th anniversary in 2020. These people were closely associated with our conferences Analytic and Algebraic Geometry. The first one is an active participant of the conferences since 2009 and the second one is a leading figure of the conferences almost from the beginning (1983). Thanks to their mathematical vigor and stimulation the conferences become more interesting and fruitful.

**Series: **Uniwersytet Łódzki

**E-ISBN-13:**978-83-8331-093-0**Print-ISBN-13:**978-83-8331-092-3**Page Count:**184**Publication Year:**2022**Language:**English

###### Wojciech Kucharz – Scientific biography

### Wojciech Kucharz – Scientific biography

(Wojciech Kucharz – Scientific biography)

**Author(s):**Kamil Rusek**Language:**English**Subject(s):**Logic**Page Range:**11-14**No. of Pages:**3

###### Tadeusz Winiarski – Scientific biography

### Tadeusz Winiarski – Scientific biography

(Tadeusz Winiarski – Scientific biography)

**Author(s):**Kamil Rusek, Marcin Dumnicki, Piotr Tworzewski**Language:**English**Subject(s):**Logic**Page Range:**16-18**No. of Pages:**3

###### Convexifying of polynomials by convex factor

### Convexifying of polynomials by convex factor

(Convexifying of polynomials by convex factor)

**Author(s):**Abdulljabar Naji Abdullah, Klaudia Rosiak, Stanisław Spodzieja**Language:**English**Subject(s):**Logic**Page Range:**21-51**No. of Pages:**31**Keywords:**Polynomial; semialgebraic set; convex function; strongly convex function; logarithmically strongly convex function; critical poin**Summary/Abstract:**W pracy podajemy nowe wyniki dotyczące "uwypuklania" wielomianów rzeczywistych, a w szczególności uogólniamy niektóre z rezultatów z pracy K. Kurdyka, S. Spodzieja, „Convexifying positive polynomials and sums of squares approximation”, SIAM J. Optim. 25 (2015), no. 4, 2512-2536. Pokazujemy też pewne zastosowania uzyskanych wyników w optymalizacji.

###### Zariski multiplicity conjecture in families of non-degenerate singularities

### Zariski multiplicity conjecture in families of non-degenerate singularities

(Zariski multiplicity conjecture in families of non-degenerate singularities)

**Author(s):**Szymon Brzostowski, Tadeusz Krasiński, Grzegorz Oleksik**Language:**English**Subject(s):**Logic**Page Range:**53-60**No. of Pages:**8**Keywords:**μ-constant; Zariski multiplicity conjecture;**Summary/Abstract:**We give a new, elementary proof of the Zariski multiplicity conjecturein μ-constant families of non-degenerate singularities.

###### On Lê’s formula in arbitrary characteristic

### On Lê’s formula in arbitrary characteristic

(On Lê’s formula in arbitrary characteristic)

**Author(s):**Evelia Rosa García Barroso, Arkadiusz Płoski**Language:**English**Subject(s):**Logic**Page Range:**61-66**No. of Pages:**6**Keywords:**milnor number; jacobian determinant**Summary/Abstract:**In this note we extend, to arbitrary characteristic, Lê's formula (Calculation of Milnor number of isolated singularity of complete intersection. Funct. Anal. Appl. 8, 127-131 (1974))

###### Lefschetz numbers and asymptotic periods

### Lefschetz numbers and asymptotic periods

(Lefschetz numbers and asymptotic periods)

**Author(s):**Karol Gryszka**Language:**English**Subject(s):**Logic**Page Range:**67-73**No. of Pages:**7**Keywords:**lefschetz number; Euler characteristic; dynamical system; asymptotic**Summary/Abstract:**In this note we prove several results linking Lefschetz numbers with asymptotic behaviour of the orbit in flows. With the aid of the Lefschetz fixed point theorem and the presence of a non-trivial limit set we prove the existence of asymptotically non-periodic orbits.

###### On the nearly free simplicial line arrangements with up to 27 lines

### On the nearly free simplicial line arrangements with up to 27 lines

(On the nearly free simplicial line arrangements with up to 27 lines)

**Author(s):**Marek Janasz**Language:**English**Subject(s):**Logic**Page Range:**75-82**No. of Pages:**8**Keywords:**hypersurface arrangements; freeness**Summary/Abstract:**In the present note we provide a complete classification of nearly free (and not free simultaneously) simplicial arrangements of d ⩽ 27 lines.

###### Realizability of some Boroczky arrangements over the rational numbers

### Realizability of some Boroczky arrangements over the rational numbers

(Realizability of some Boroczky arrangements over the rational numbers)

**Author(s):**Marek Janasz, Magdalena Lampa-Baczyńska, Daniel Wójcik**Language:**English**Subject(s):**Logic**Page Range:**83-93**No. of Pages:**22**Keywords:**boroczky arrangements; Grobner basis; parametrization; points matrix**Summary/Abstract:**In this paper, we study the parameter spaces for Böröoczky arrangements Bn of n lines, where n < 12. We prove that up to n = 12, there exist only one arrangement nonrealizable over the rational numbers, that is B11.

###### Effective proof of Guseĭn-Zade theorem that branches may be deformed with jump one

### Effective proof of Guseĭn-Zade theorem that branches may be deformed with jump one

(Effective proof of Guseĭn-Zade theorem that branches may be deformed with jump one)

**Author(s):**Andrzej Lenarcik, Mateusz Masternak**Language:**English**Subject(s):**Logic**Page Range:**95-119**No. of Pages:**25**Keywords:**plane curve singularity; Milnor number; deformations of singularities; Newton algorithm**Summary/Abstract:**W pracy podajemy efektywny dowód twierdzenia Gusein-Zade, że osobliwe nierozkładalne lokalne krzywe płaskie (gałęzie) mogą być deformowane ze skokiem liczby Milnora równym jeden. W dowodzie korzystamy z wersji twierdzenia Kouchnirenki dostosowanej do algorytmu Newtona w wersji Cano przedstawionego w pracy A.Lenarcik „Polar quotients of plane curve and the Newton algorithm, Kodai Math. J. 27 (2004), 336-353.

###### Real Nullstellensatz and sums of squares

### Real Nullstellensatz and sums of squares

(Real Nullstellensatz and sums of squares)

**Author(s):**Maria Michalska**Language:**English**Subject(s):**Logic**Page Range:**121-136**No. of Pages:**16**Keywords:**Nullstellensatz; sums of squares; Artin-Schreier; Hilbert’s 17th Problem**Summary/Abstract:**In this paper we highlight the foundational principles of sums of squares in the study of Real Algebraic Geometry. To this aim the article is designed as mainly a self-contained presentation of a variation of the standard proof of Real Nullstellensatz, the only relevant omission being the (long) proof of the Tarski-Seidenberg theorem. On the way we see how the theory follows closely developments in algebra and model theory due to Artin and Schreier. This allows us to present on the way Artin’s solution to Hilbert’s 17th Problem: whether positive polynomials are sums of squares. These notes are intended to be accessible to math students of any level.

###### Some notes on the Lê numbers in the family of line singularities

### Some notes on the Lê numbers in the family of line singularities

(Some notes on the Lê numbers in the family of line singularities)

**Author(s):**Grzegorz Oleksik, Adam Różycki**Language:**English**Subject(s):**Logic**Page Range:**137-146**No. of Pages:**10**Keywords:**Jump of Le numbers; Non-isolated hypersurface singularity; Le numbers; Newton diagram; Modified Newton numbers; Iomdine-Le-Massey formula**Summary/Abstract:**In this paper we introduce the jumps of the Lê numbers of nonisolated singularity f in the family of line deformations. Moreover, we prove the existence of a deformation of a non-degenerate singularity f such that the first Lê number is constant and the zeroth Lê number jumps down to zero. We also give estimations of the Lê numbers when the critical locus is one-dimensional. These give a version of the celebrated theorem of A. G. Kouchnirenko in this case.

###### Lectures on polynomial equations: Max Noether’s Fundamental Theorem, The Jacobi Formula and Bézout’s Theorem

### Lectures on polynomial equations: Max Noether’s Fundamental Theorem, The Jacobi Formula and Bézout’s Theorem

(Lectures on polynomial equations: Max Noether’s Fundamental Theorem, The Jacobi Formula and Bézout’s Theorem)

**Author(s):**Arkadiusz Płoski**Language:**Polish**Subject(s):**Logic**Page Range:**147-162**No. of Pages:**16**Keywords:**Jacobi Formula; Max Noether’s Fundamental Theorem; Bézout’s Theorem**Summary/Abstract:**Using some commutative algebra we prove Max Noether’s Theorem, the Jacobi Formula and B´ezout’s Theorem for systems of polynomial equations defining transversal hypersurfaces without common points at infinity.

###### An invitation to the positivity and geometry of algebraic cycles

### An invitation to the positivity and geometry of algebraic cycles

(An invitation to the positivity and geometry of algebraic cycles)

**Author(s):**Justyna Szpond**Language:**English**Subject(s):**Logic**Page Range:**147-162**No. of Pages:**11**Keywords:**ACM subvarieties; algebraic cycles; Hartshorne conjecture; mobility count; space curves; postulation problems for cycles**Summary/Abstract:**The purpose of this work is an introduction and overview of geometric and numeric properties of algebraic cycles in smooth projective varieties. We recall or propose several problems, which we consider worth to study. We are mainly interested in, but do not restrict our story to, codimension 2 cycles in projective spaces. These are points in P^2, curves in P^3, surfaces in P^4 and so on.

###### An estimation of the jump of the Milnor number of plane curve singularities

### An estimation of the jump of the Milnor number of plane curve singularities

(An estimation of the jump of the Milnor number of plane curve singularities)

**Author(s):**Aleksandra Zakrzewska**Language:**English**Subject(s):**Logic**Page Range:**175-183**No. of Pages:**9**Keywords:**milnor number; plane curve singularities; Enriques diagram**Summary/Abstract:**The jump of the Milnor number of an isolated singularity f₀ is the minimal non-zero difference between the Milnor numbers of f₀ and one of its deformations fs. We estimate the jump using the Enriques diagram of f₀.