 is the V number of monomial ideal. Basically, this is a very recent topic. So, the contents are this that I will give some introduction and motivations of the of my talk and then we need some preliminaries and then I will talk about the V number of monomial ideals via its polarization and then I will talk about bounds of regularity and induced matching number by the V number and then some open problem. So, first let us first give the definition of the V number. Let i be a proper graded ideal of a polynomial ring and we can take the standard grading. Then the V number of i denoted by V i is defined as this that is the minimum degree of an homogeneous polynomial f for which i colon f is a prime ideal because we know for a Noetherian ring if we take an ideal then its associated primes are of the form i colon f that is i colon some polynomial. So, for an ideal actually the ideal in a polynomial ring the associated prime will be of the form i colon f and we have to take the minimum degree of such f for which i colon f gives a prime ideal. So, that is the V number. So, we can locally define the V number for each prime ideal just like for each prime ideal p belongs to for each associated prime of i we can locally define the V number as V p i such that the minimum degree of f for which i colon f is equal to that p. So, obviously, the V number will be minimum of V p i. So, this is some motivation why we study V number or where it comes actually. The V number of i was introduced as an invariant of the graded ideal i during the study of Riedmuller type codes mainly in coding theory and this invariant investigate the asymptotic behavior of the minimum distance function of projective Riedmuller type codes. And later it was seen that from a geometrical point of view which is a very in the study of 1993. So, from a geometrical point of view the local V number extends the notion of degree of a point in a finite set of projective points. And also the in this invariant can be used to give some bounds of some other invariants like Kastal-Nurban-Mamperegularity or induced matching number. So, we need some preliminaries. So, mainly we are talking about the V number of monomial ideals. So, first Zaramillo and Villarreal started the study for V number of edge ideals that is for the square free monomial ideals. So, first give the definition of clutter we know also clutter or simple hypergraph. That is a clutter C is a pair of two sets. One is a vertex set, one is the edge set where V c is called the vertex set is a collection of and E c is a collection of subset of V c such that no two elements contain each other. And that is the elements of V c is called the edge. And a simple graph is an example of a clutter or simple hypergraph. So, why we define clutter? Because we know that any square free monomial ideal can be associated with a clutter. So, the set of square free monomial ideal and the set of clutters are in one to one correspondence. So, for an edge in a clutter we define the monomial x a. We define the monomial x a like this. So, for a set of for a subset of vertex set we can define the monomial x a like the product of the variables. And then we take all x e where e belongs to E c that is for all edges we are taking the x e and the ideal generated by this. So, this is a square free monomial ideal and this is called the edge ideal of the clutter. And we know that what we say that square free monomial ideals and clutters are one to one correspondence. Now, I am talking about some preliminaries of the combinatorics such that a subset A of the vertex set subset of a vertex set is said to be stable or independent if there exist no edge contained in A. There is no edge contained in A that is called the independent set. And what is the neighbor of a set? So, neighbor of a set means neighbor of a independent set. So, we have to take an independent set and we have to collect those vertices such that the x i union that those vertices x i such that x i union that independent set contains some edges. So, that is called the neighbor set. And if we union A with the neighbor of A then we it is seen as a close neighbor of A. A subset c of v c that is a vertex set is called the vertex cover of c if e intersection c is non empty for any e belongs to e c. And f c denotes the collection of all maximal stable sets of c or maximal independent sets of c. So, it is very well known that for a square free monomial ideal the associated primes are related with the minimal vertex cover of the clutter or simple hypergraph. So, here we are talking about the maximal stable sets. So, if we go to the complement of maximal stable set it is a minimal vertex cover. So, the complement of a maximal stable set always give a minimal vertex cover or an associated prime we can say. And a c denotes the collection of those independent sets such that its neighbor is a minimal vertex cover. So, why we define this because in the first paper they gave the combinatorial description of the v number that is Zaramillo and Villarreal gave the combinatorial description of the v number for square free monomial ideal. So, how they gave this is written here that suppose i equal to i c be the edge ideal of a clutter and i is not a prime because for the prime case v number will be 0 ok we can take any constant or something. Then f c is contained in a c and v number is the minimum of the cumulative a such that a belongs to a c. So, what is actually a? So, if I just draw a graph for normal. So, here we can see x 1 and x 5 is a maximal independent sets. So, its complement that is or its neighbor x 2, x 3, x 4 is a minimal vertex cover. So, if I take the edge ideal of this graph suppose G and colon width with the monomial x 1, x 5 then actually we will get the prime ideal x 2, x 3, x 4. So, the degree of x 1, x 5 is 2. So, we can say that the v number is less equal to 2, but if you see if we do not take x 5 then x 1 itself is a independent sets and the neighbor of x 1 is also x 2, x 3, x 4. So, this is also like this. So, here x 1 does the job. So, we can say that the v number of this ideal is 1. So, from this we can say because this is not a prime ideal. So, v number cannot be 0. So, this is a combinatorial description for the v number for square free monomial ideal. Now, we are talking about some the induced matching of graphs. So, first call the what is matching. So, let G be a graph a matching is a subset of the edge set such that no two edges of M shares a common vertex. And a matching is called an induced matching in G if the induced subgraph on the vertex set union of EI that is M the edges of in M contains only M as the edge set. So, such like if we take this graph this edge and this edge forms a matching, but this is not an induced matching why because in the induced subgraph there is an edge. So, if we remove this edge and consider this graph then this two will be an induced matching. And it is an well known result that the induced match for a edge ideal induced matching is less equal to the regularity of the ideal. That is why it is very important. And the cardinality of the maximum induced matching in G is known as the induced matching number and we will denote it by IMG. Now, talk about the v number of monomial ideal via polarization because polarization is such a technique such that we can extend the monomial ideal into the square phi monomial ideal and we can get some results of properties or invariance very easily. So, this is the definition of polarization I think I can skip this because we know about that. So, mainly this polarization is nothing but the corresponding to a monomial ideal in an extended ring in an extended polynomial ring we can see it as a square phi monomial ideal and then we can relate some invariance or some properties such like Cohen-Makall property, regularity or dimension or depth. Let I be an arbitrary monomial ideal. If there exists a prime ideal associated prime ideal of I pole, I pole denotes the polarization of I. If there exists a prime ideal of associated prime of sorry if there exists an associated prime of I pole such that XS1 BS1 to XSK BSK such that there is no embedded prime of I containing XS1 to XSK. So, it is a well known result that if we take an associated prime of the polarization of a monomial ideal then it would look like this that is XS1 BS1 to XSK BSK but XS1 XS2 XSK is an associated prime of I. So, this will be an associated prime of I and suppose if we see in I there is no embedded prime containing this ideal then we can say the V number of the polarization is equal to Vp I pole and sorry not like that the V number of the polarization attend for this prime ideal this P then we can say V number of I is equal to V number of the polarization. So, when we are talking about the V number, V number can be attained for a prime ideal. So, it may attend for the multiple prime ideal. So, if there exists some prime so, if there exists some prime ideal like this for which the V number is attained and the corresponding prime ideal is contained no embedded prime in I then V number of I is equal to the polarization and in general we get the V number of the polarization is less equal to V number of I and if I has no embedded prime then V number of I pole is equal to V I for the unmixed case or Cohen Macaulay case also we can say that V number of I pole is equal to V number of I. So, the converse of corollary 2.2 is not true because just like V number of I pole can be V number of we can be equal to V number of I, but I may have some embedded prime and it is clear from our example here. So, I is a monomial ideal and this is the primary decomposition of I. So, here the associated primes of I are x 2, x 3, x 1, x 3, x 1, x 2, x 1, x 2, x 3. So, x 1, x 2, x 3 is not an embedded prime of I, but there is no associated prime of I containing x 1, x 2, x 3 and by the algorithm given in 4 we can compute that V number of I and V number of I polarization is 3 and we can actually see that I colon x 1, x 2, x 3 is nothing but the ideal generated by x 1, x 2, x 3. So, here V number is attained for this prime ideal and this does not contain any embedded prime. This is an embedded prime of I, but there is no embedded prime of I containing x 1, x 2, x 3. So, from our theorem, so this justifies the V number of I is equal to V number of I pole. And also I pole we can see like this that is this will be an associated prime. We also proved that so in the study of polynomial ring the this exact sequence is very important whenever we study a monomial ideal or something. So, it is very important to see the relation between the V number of I, V number of I colon F and V number of I comma F. So, here we got the result. So, let I be a monomial ideal and F be a monomial such that F does not belongs to I. Then we can say that V number of I is less equal to V number of I colon F plus degree of F. Then for a graph G, we prove that V number of the edge ideal of G is less equal to the vertex covering number of the graph. The additivity of the V number for monomial ideal we have also proved that is if I 1 is ideal in a polynomial ring and I 2 is a ideal in a different polynomial ring, then we have V number of I 1 R plus I 2 R in the extended polynomial ring where R is the tensor of R 1 and R 2, V number of I 1 plus V number of I 2. So, this is proved for the monomial ideal and somewhat recently we have proved for the binomial edge ideal, but for the arbitrary graded ideal we do not know the V number will be additive or not. So, this is an interesting thing to see. For the complaining decision ideal, we see that V number of I is nothing but the degree of xi xi i minus k and which is equal to the regularity. So, for complaining decision monomial ideal, V number is equal to the regularity of Arcauchin time and these are some property what I say about this exact sequence to see how they related. So, here we give some relation such that i equal to i c be an edge ideal of a clatter that is i is an square free monomial ideal. And suppose xi does not appear in the edge set that is xi does not appear in the generating set but as a single variable then V number of I is less equal to V number of I colon xi plus 1. Also for some xi we get V number of I colon xi is less equal to Vi. Also if Vi is greater equal to 2 then V number of I colon xi is less than V number of I for some xi. And also V number of I c minus xi that means V number of I comma xi is less equal to V number of I c for some xi belongs to V c. Next we talk about the bounds of regularity and induced matching number by the V number. So, in this section the main result of is the following that if g is a bipartite graph or c 4 c 5 free vertex g composable graph which also include the chordal graph or whisker graph. For in this case we see that we prove that the V number of I g is less equal to the induced matching number of g less equal to the regularity of R quotient I g. And also for the any positive integer n we have a graph g such that regularity of R quotient I g minus V number of I g is n. That is for every positive integer we can construct a graph such that this hold. So, we can say that regularity is can be arbitrarily larger than the V number. It is a definition of the Alexander dual. So, Alexander dual is nothing but if we take an square free monomial ideal we have to go to the primary decomposition and correspond to primary decomposition we can construct a monomial ideal which is known as also the cover ideal. And then ideal is known as the Alexander dual because if we dual of this ideal also becomes the main ideal that is I. So, this is called the cover ideal or Alexander dual of a square free monomial ideal. And we see that if let C be a clutter that cannot be written as a union of two disjoint clutters or we can say any clutter also. If V number of the Alexander dual is less equal to the regularity of R quotient Alexander dual of I plus 1 and V number of Alexander dual of I is greater equal to the vertex covering number of C plus 1 then I cannot be Cohen-McCullough. So, this is an relation of the Cohen-McCulloughness and V number of square free monomial ideal. And Zaramillo and Villarreal give propose some open problem whether V number is less equal to regularity plus regularity of R quotient I plus 1 or not for any square free monomial ideal. And we give some example where this does not hold and also later recently C1, use of C1 proved that for every integer K greater than 1 there exists a connected graph HK such that V number of IHK is less equal to regularity of IHK plus 1. We have proved that regularity can be arbitrary larger than V number and here he proved that V number can be arbitrary larger than regularity. So, this is an open problem this is a problem that classification of some classes of connected graph G for which this holds. We proved for the bipartite one C4, C5 free vertex decomposable and also for whisker graph. So, all other classes there or not some good classes for which this holds. And that and also we notice that for the Cohen-McCullough case we from our result we can see that the V number is less equal to the depth and the V number is greater equal to the dimension minus depth which is nothing but 0 actually in the Cohen-McCullough case. And also in some example or many example when the Cohen-McCulloughness is not necessary we see that this inequality holds. So, we put the following question for a square free monomial ideal does V number is less equal to the depth or not or V number is greater equal to dimension minus depth or not. Still we do not know about this see the and this is the paper from.